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Angles, arc lengths and trig functions

Angles, arc lengths and trig functions on the SAT test

SAT Subscore: Additional topics in math

Angles, arc length and trig functions subject includes 3 topics:
Calculations of angles in radians.
Calculation of arc lengths and sector areas in radians.
Calculation of sine, cosine and tangent in radians.

Before learning this subject learn the pages about circle theorems and right triangle trigonometry.

A radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. To find the number of radians in a circle, we need to divide the length of the circumference by the radius: 2πr/r=2π, meaning that the number of radians of arc in a circle is 2π.

The relationship between radian and degree measures:
2π radians is equal to 360 degrees, meaning that one radian is equal to 360/2π=180/π=180/3.14≈57 degrees.

Radian measure   =    degree measure
_________________        _________________
            π                                 180°

The relationship between central angle in radians, arc length and sector area:
central angle    =         arc length              =     sector area
_____________       ____________________         ____________
        2π                   circle circumference           circle area

Special right triangles in circles:
In these questions we are given a circle which center is located at the axis intersection point (0,0).

Special right triangles are right triangles whose sides are in a particular ratio.
Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

The radian measures of angles of special right triangles are:
30° angle radian measure π/6; 45° angle radian measure π/4, 60° angle radian measure π/3; 90° angle radian measure: π/2.

We can draw a right triangle from any point on the circle so that the hypotenuse is equal to the radius of the circle and the sides of the triangle are equal to x and y coordinates of the point.

We can check the ratios between the sides of the triangle to see if the triangles are special right triangles. If the triangles are special right triangles, we know the values of their angles.

Calculating trigonometric functions (sin, cos and tan) with radian angle measures in right triangles in circles:
We calculate trigonometric functions under the assumption of unit circle, meaning that the radius is equal to 1. Since the hypotenuse is equal to the radius, we know that the hypotenuse is equal to 1.

The trigonometric functions values are:
sin(A)=opposite/hypotenuse= opposite/1=opposite.
cos(A)=adjacent/hypotenuse= adjacent/1=adjacent.
tan(A)=opposite/adjacent.

Continue reading this page for detailed explanations and examples.

The relationship between radian and degree measures

Degrees in a circle: The number of degrees of arc in a circle is 360.

Radians in a circle: To find the number of radians in a circle, we need to divide the length of the circumference by the radius: 2πr/r=2π, meaning that the number of radians of arc in a circle is 2π.

Note that the underlined sentences above are provided at the beginning of each SAT math section.

Therefore, the connection is that 2π radians is equal to 360 degrees, meaning that one radian is equal to 360/2π=180/π=180/3.14≈57 degrees.

The figure below presents the connection between radian and degree measures.

A radian is defined as the angle subtended from the center of a circle (marked in red) which intercepts an arc equal in length to the radius of the circle (the radii and the equal arc are marked in blue).

The connection between radians and degrees

Since 2π radians is equal to 360 degrees, we can calculate radian measure given degree measure or calculate degree measure given radian measure using the following ratios:

Radian measure   =    degree measure
________________        _________________
           2π                               360°

We can simplify the proportion getting:

Radian measure   =    degree measure
_________________        _________________
             π                                 180°

Consider the following example:

Covert 100° to radians.

Radian measure   =    degree measure
________________         ________________
             π                                 180°

Represent radian measure by the variable x.

x    =    100
__         ____
π          180

180x=100π
x=100/180π
x=0.55π=0.55*3.14=1.74 radians
100 degrees are equal to 0.55π radians.

Consider the following example:

Covert 3.5 radians to degrees.

Radian measure   =    degree measure
________________         ________________
             π                                 180°

Represent degree measure by the variable x.

3.5   =    x
___        ___
 π          180

πx=180*3.5
x=180*3.5/π
x=200°
3.5 radians are equal to 200 degrees.

We can also calculate without using the proportion using the fact that 1 radian is equal to 57 degrees:
3.5*57=200

Calculating angles, arc length and sector areas with radians

We can measure arc length and sector areas with radians instead of degrees.

We know that the relationship between central angle in degrees, arc length and sector area is given by the following ratios:

central angle    =            arc length              =     sector area
_____________        _____________________          ____________
       360°                  circle circumference            circle area

We also know that the number of radians of arc in a circle is 2π, therefore we can substitute 360 degrees by 2π:

The relationship between central angle in radians, arc length and sector area is given by the following ratios:

central angle    =         arc length              =     sector area
_____________       ____________________         ____________
          2π                circle circumference           circle area

Consider the following example:

The central angle of a circle is equal to 0.5π, the circumference of the circle is equal to 10 centimeters.

What is the measure of the arc formed by this angle?

What is the measure of the sector area formed by this angle?

Calculating the arc length:

central angle    =         arc length
_____________        ______________________
         2π                    circle circumference

Represent arc length by the variable x and plug the given data into the ratios equation:

0.5 π  =   x
____     ___
2π        10

x=0.5*10/2
x=2.5
The arc length is 2.5 centimeters.

Checking the answer:
0.5 π  = 2.5
____     ___
2π        10

0.5/2=2.5/10
1/4=1/4

Calculating the sector area:
The circle area formula is A=πr2.
A=π*102
A=100π
A=100*3.14=314

central angle    =    sector area
_____________        __________
2π                      circle area

Represent sector area by the variable x and plug the given data into the ratios equation:

0.5π  =     x
____      _____
2π         100π

x=0.5*100π/2
x=25π
The sector area is 25π.

Checking the answer:
0.5π  =   25π
____      _____
2π         100π

0.5/2=25/100
1/4=1/4

Special right triangles in circles

In these questions we are given a circle which center is located at the axis intersection point (0,0)

We can draw a right triangle from any point on the circle so that the hypotenuse is equal to the radius of the circle and the sides of the triangle are equal to x and y coordinates of the point.

We can check the ratios between the sides of the triangle to see if the triangles are special right triangles. If the triangles are special right triangles, we know the values of their angles. 

Special right triangles measures

Special right triangles are right triangles whose sides are in a particular ratio.
Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.
Special right triangles with their side sizes length are given at the beginning of each SAT section.

30°, 60°, 90° triangle:
In a 30°, 60°, 90° right triangle the side opposite the 30° angle is half the length of the hypotenuse and the side opposite to 60° angle is equal to the length of a side opposite to 30° angle multiplied by √3.
In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

 45°, 45°, 90° triangle:
In a 45°, 45°, 90° right triangle the sides opposite the 45° angles are equal and the hypotenuse is equal to the side opposite to 45° angle multiplied by √2.
In 45°, 45°, 90° triangle the sides are s, s and s√2.

The following graphs present the special right triangles with the side sizes length.

Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

Radian measures of angles of special right triangles

30° angle radian measure: π/6

Since 2π radians is equal to 360 degrees we get:

Radian measure   =    degree measure
________________         ________________
             π                                180°

x   =  30
__     ___
π      180

x=30π/180
x=π/6

45° angle radian measure: π/4

Since 2π radians is equal to 360 degrees we get:

Radian measure   =    degree measure
________________         ________________
             π                                 180°

Representing radian measure by the variable x and plugging the degree measure into the proportion:

x  = 45
__   ___
π    180

x=45π/180
x=π/4

Since 30*1.5=45 we can also multiply the radian measure of 30° angle by 1.5 getting:
1.5*π/6= π/4.

60° angle radian measure: π/3

Since 2π radians is equal to 360 degrees we get:

Radian measure   =    degree measure
________________         ________________
             π                                 180°

Representing radian measure by the variable x and plugging the degree measure into the proportion:

x  = 60
__   ___
π    180

x=60π/180
x=π/3

Since 30*2=60 we can also multiply the radian measure of 30° angle by 2 getting:
2*π/6= π/3.

90° angle radian measure: π/2

Since 2π radians is equal to 360 degrees we get:

Radian measure   =    degree measure
________________         ________________
              π                               180°

Representing radian measure by the variable x and plugging the degree measure into the proportion:

x  = 90
__   ___
π    180

x=90π/180
x=π/2

Since 30*3=90 we can also multiply the radian measure of 30° angle by 3 getting:
3*π/6= π/2.

The following figures present special right triangles with their side lengths and angles in degrees and radians (the radian measures are marked in red).

Special right triangles radian and degree values

Calculating side lengths and radian angle measures in special right triangles in circles

We are given a circle which center is located at the axis intersection point (0,0)

We can draw a right triangle from any point on the circle so that the hypotenuse is equal to the radius of the circle and the sides of the triangle are equal to x and y coordinates of the point.

We can check the ratios between the sides of the triangle to see if the triangles are special right triangles. If the triangles are special right triangles, we know the values of their angles. 

Consider the following example:

The points coordinates are A(3,3) and b(-4,4/√3). Both points are located on a circle and the center of the circle is located at the axis intersection point (0,0).

What is the size of the angle BOD?

What is the size of the angle AOC?

Finding side lengths and radian angle measures in special right triangles in circles

The figure above presents 2 points A and B that are located on a circle. The center of the circle is located at the axis intersection point (0,0). The coordinates of point A are (3,3) and the coordinates of point B are (-4,4/√3).

Drawing right triangles from points on a circle and calculating side length from the coordinates of the points:

The lines AO and BO are radii of the circle. 

The line AC in drawn from point A to create a right triangle ACO, so that the angle ACO is equal to 90°. Since the angle ACO is equal to 90°, the length of the side AC is equal to y coordinate of point A so that AC=3. In addition, the length of the side CO is equal to x coordinate of point A so that CO=3.

The line BD in drawn from point A to create a right triangle BDO, so that the angle BDO is equal to 90°. Since the angle BDO is equal to 90°, the length of the side BD is equal to the y coordinate of point B so that BD=4/√3. In addition, the length of the side DO is equal to the absolute value of the x coordinate of point B so that DO=|-4|=4.

Calculating the angles of special right triangles:

In the triangle ACO the side lengths are AO=CO=3, therefore the triangle ACO is an isosceles triangle. An isosceles triangle is a special right triangle and we know its angles measures are 45°,45° and 90° and the radian measures are π/4, π/4 and π/2.

Note that we can also calculate the angle measures: Since the triangle ACO is an isosceles triangle, the angles CAO and AOC are equal. Since the angle ACO is equal to 90° and the sum of the angles in a triangle is 180°, the angles CAO and AOC are equal to 45°.

In the triangle BDO the side lengths are BD=4/√3 and DO=4, therefore the triangle ACO is a special right triangle and its angles measures are 30°,60° and 90° and the radian measures are π/6, π/3 and π/2. Therefore, the angle BOD=30°= π/6 and the angle DBO=60°=π/3.

Calculating trigonometric functions (sin, cos and tan) with radian angle measures in right triangles in circles

Special right triangles with their side sizes length are given at the beginning of each SAT section.

The following figure presents 2 special right triangles and their angles in degrees, like given in the SAT (note that the radian measures are not given).

Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

We calculate trigonometric functions under the assumption of unit circle, meaning that the radius is equal to 1. Since the hypotenuse is equal to the radius, we know that the hypotenuse is equal to 1.

Remember the trigonometric functions values:
sin(A)=opposite/hypotenuse= opposite/1=opposite.
cos(A)=adjacent/hypotenuse= adjacent/1=adjacent.
tan(A)=opposite/adjacent.

To learn more about trigonometric functions go to right triangle trigonometry page.

Calculating the radian angles measures of triangles

At the beginning of each SAT math section, it is given that:
The number of degrees of arc in a circle is 360.
The number of radians of arc in a circle is 2π.

Therefore, we know that 360°=2π and π=180°.
The angle of 30°: 30=360/12=2π/12=π/6.
The angle of 45°: 45=360/8=2π/8=π/4.
The angle of 60°: 60=360/6=2π/6=π/3.
The angle of 90°: 90=360/4=2π/4=π/2.
The angle of 120°: 120=360/3=2π/3.
The angle of 135°: 135=360*3/8=2π*3/8=3π/4.
The angle of 180°: 180=360/2=2π/2=π.

Special right triangle with 30°, 60°, 90° angles

Calculating the side lengths of the triangle:
We know the side lengths from the beginning of each math SAT section: x, 2x and x√3. Since the hypotenuse is equal to 1 (unit circle) we know that 2x=1 and x=1/2. Therefore, the sides are 1/2, √3/2 and 1.

Calculating the sine, cosine and tangent of the angle of π/6 radians:
We need to look at the given 30°, 60°, 90° special right triangle, we saw that its sides are 1/2, √3/2 and 1.
sin (π/6)=opposite/1=(1/2)/1=1/2.
cos (π/6)=adjacent/1=(√3/2)/1=√3/2.
tan (π/6)=opposite/adjacent=(1/2)/(√3/2)=1/√3 multiply by √3/√3 getting (1*√3)/(√3*√3)= √3/√9=√3/3.

Calculating the sine, cosine and tangent of the angle of π/3 radians:
We need to look at the given 30°, 60°, 90° special right triangle, we saw that its sides are 1/2, √3/2 and 1.
sin(π/3)=opposite/1=√3/2.
cos(π/3)=adjacent/1=1/2.
tan(π/3)=opposite/adjacent=(√3/2)/(1/2)= √3.

Special right triangle with 45°, 45°, 90° angles

Calculating the side lengths of the triangle:

We know the side lengths from the beginning of each math SAT section: s, s and s√2. Since the hypotenuse is equal to 1 (unit circle) we know that s√2=1 and s=1/√2. Therefore, the sides are 1/2, 1/2 and 1.

Calculating the sine, cosine and tangent of the angle of π/4 radians:
We need to look at the given 45°, 45°, 90° special right triangle.
sin(π/4)=opposite/1=1/√2 multiplying by √2/√2 getting (1*√2)/( √2*√2)=√2/2.
cos(π/4)=adjacent/1=1/√2 multiplying by √2/√2 getting (1*√2)/( √2*√2)=√2/2.
tan(π/4)=opposite/adjacent=(√2/2)/(√2/2)=1.

Angle measures of 0 radians and π/2 radians

Calculating the sine, cosine and tangent of the angle of 0 radians:
sin(0)=opposite/1: If the angle is close to 0, the side that is opposite to the angle is also close to 0, therefore sin(0)=0/1=0.
cos(0)=adjacent/1= If the angle is close to 0, the side that is adjacent to the angle is almost equal to the hypotenuse which is equal to 1, therefore cos(0)=1/1=1.
tan(0)=opposite/adjacent= If the angle is close to 0, the side that is opposite to the angle is also close to 0 and the side that is adjacent to the angle is equal to the hypotenuse which is equal to 1 therefore tan(0)=0/1=0.

Calculating the sine, cosine and tangent of the angle of π/2 radians:
We know that 2π=360°, therefore π/2=90°.
sin(π/2)=opposite/1=If the angle is close to 90°, the side that is opposite to the angle is almost equal to the hypotenuse which is equal to 1, therefore sin(π/2)=1/1=1.
cos(π/2)=adjacent/1= If the angle is close to 90°, the side that is adjacent to the angle is close to 0, therefore cos(π/2)=0/1=0.
tan(π/2)=opposite/adjacent= If the angle is close to 90°, the side that is opposite to the angle is 1 and the side that is adjacent to the angle is 0. We can’t divide by 0, therefore tan(π/2) is not defined.

Angle measures bigger than 90°:

We can convert these angles to angles smaller than 90° using 2 formulas:
sin(α)=sin(180-α) or with radians: sin(α)=sin(π-α).
cos(α)=-cos(180-α) or with radians: cos(α)=-cos(π-α).
We also know that tan(α)=sin(α)/cos(α)

Calculating the sine, cosine and tangent of the angle of 2π/3 radians:
We know that 2π=360°, therefore 2π/3=360°/3=120°>90°.
sin(2π/3)= sin(π-2π/3)=sin{(3π-2π)/3}=sin(π/3), we found that sin(π/3)=√3/2.
cos(2π/3)=-cos(π-2π/3)= -cos{(3π-2π)/3}=-cos(π/3), we found that cos(π/3)=1/2, therefore -cos(π/3)=-1/2.
tan(2π/3)= sin(2π/3)/cos(2π/3)=(√3/2)/(-1/2)=-(√3/2)*2=-√3.

Calculating the sine, cosine and tangent of the angle of 3π/4 radians:
We know that 2π=360°, therefore π=180° and 3π/4=180°*3/4=135°>90°.
sin(3π/4)= sin(π-3π/4)=sin{(4π-3π)/4}=sin(π/4), we found that sin(π/4)=√2/2.
cos(3π/4)=-cos(π-3π/4)=-cos{(4π-3π)/4}=-cos(π/4), we found thar cos(π/4)=√2/2, therefore -cos(π/4)=-√2/2.
tan(3π/4)=sin(3π/4)/cos(3π/4)=(√2/2)/(-√2/2)=-1.

Calculating the sine, cosine and tangent of the angle of π radians:
Since 2π=360°, the angle of π radians is equal to 180°.
sin(π)=sin(π-π)=sin(0)=0.
cos(π)=-cos(π-π)=-cos(0)=-1.
tan(π)=sin(π)/cos(π)=0/-1=0.

Right triangle trigonometry and word problems

Right triangle trigonometry and word problems on the SAT test

SAT Subscore: Additional topics in math

Right triangle is a triangle with a right angle (equal to 90°). The side opposite the right angle (the longest side of the right triangle) is called a hypotenuse.

Right triangle trigonometry and right triangle word problems require calculating side lengths and angle measures in right triangles.

Pythagorean theorem:

Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides: a2 + b2 = c2

Pythagorean triples are combinations of side lengths a, b and c that satisfy the Pythagorean theorem. If you remember the triples values, you know the size of the third side without the need to calculate it. The most common Pythagorean triples are: 3, 4 and 5;    5, 12 and 13;    7, 24 and 25.  

Trigonometric ratios (sine, cosine and tangent):

Trigonometric ratios (functions) represent connections between angle degrees and side lengths in a right triangle:

  • The sine of an angle (sin) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, to the length of the hypotenuse. 
  • The cosine of an angle (cos) in a right triangle is defined as the ratio of the length of the side that is adjacent to the angle, to the length of the hypotenuse. 
  • The tangent of an angle (tan) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, side that is adjacent to the angle. 

Complementary angles are two angles with the sum of 90 degrees. Sine of an angle (α) in a right triangle is equal to cosine of its complementary angle (90-α). 

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio therefore they also have similar sine, cosine and tangent.

Special right triangles:

Special right triangles are right triangles whose sides are in a particular ratio. 

  • In a 30°, 60°, 90° right triangle the side opposite the 30° angle is half the length of the hypotenuse and the side opposite to 60° angle is equal to the length of a side opposite to 30° angle multiplied by √3.
    In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.
  • In a 45°, 45°, 90° right triangle the sides opposite the 45° angles are equal and the hypotenuse is equal to the side opposite to 45° angle multiplied by √2.
    In 45°, 45°, 90° triangle the sides are s, s and s√2.

Continue reading this page for detailed explanations and examples.

Pythagorean theorem in right triangles

Pythagorean theorem states that the square of the length of the hypotenuse of a right triangle equals the sum of the squares of the lengths of the other two sides. (The hypotenuse is the side opposite the right angle).

The formula of the Pythagorean theorem is a2 + b2 = c2. This formula is provided at the beginning of each math section followed by a diagram.  

The following figure presents a right triangle and the Pythagorean theorem a2 + b2 = c2.

Pythagorean theorem in a triangle

Calculating side lengths with Pythagorean theorem

If we are given the values of two sides of a right triangle, we can calculate the value of the hypotenuse with the Pythagorean theorem.

If we are given the values of one side and the hypotenuse of a right triangle, we can calculate the value of the second side with the Pythagorean theorem.

Consider the following example:

The length of the sides of a right triangle are 3 centimeters and 4 centimeters.

What is the length of the hypotenuse?

The formula of the Pythagorean theorem is a2+b2=c2.
a=3, b=4, c=?
32 + 42 = c2
c2=9+16
c2=25
c=5
The length of the hypotenuse is 5 centimeters.

Checking the answer:
32+42=52
9+16=25
25=25

Consider the following example:

The length of a side of a right triangle is 4 centimeters and the length of the hypotenuse is 8 centimeters.

What is the length of the other side?

The formula of the Pythagorean theorem is a2+b2=c2.
a=4, b=?, c=8
42+b2=82
16+b2=64
b2=64-16
b2=48
b=√48
b=√(4*12)
b=√4*√12
b=2√12=6.93

Checking the answer:
a2+b2=c2
42+(2√12)2=82
16+4*12=64
16+48=64
64=64

Pythagorean triples in a right triangle

Pythagorean triples are combinations of side lengths a, b and c that satisfy the Pythagorean theorem. If you remember the triples values, you know the size of the third side without the need to calculate it.

The most common Pythagorean triple is 3, 4 and 5.
Additional Pythagorean triples are:
5, 12 and 13
7, 24 and 25  

Note that each multiplication of the triple also satisfies the Pythagorean theorem.
For example: The triple 9, 12 and 15 is the triple 3, 4 and 5 multiplies by 3. Therefore, the triple 9, 12 and 15 is a Pythagorean triple.
We can check this statement by plugging into the formula of the Pythagorean theorem:
a2+b2=c2
92+122=152
81+144=225
225=225

Consider the following example:

The length of a side of a right triangle is 7 centimeters and the length of the hypotenuse is 25 centimeters.

What is the length of the other side?

The triple 7, 24 and 25 is a Pythagorean triple, therefore the length of the second side is 24 centimeters.

Checking the answer:
a2+b2=c2
72+b2=252
b2=252-72
b2=625-49
b2=576
b=24
The length of the other side is 24 centimeters.

Consider the following example:

The length of the sides of a right triangle are 10 centimeters and 24 centimeters.

What is the length of the hypotenuse?

The triple 5, 12 and 13 is a Pythagorean triple, the given sides 10 and 24 are the triple sides 5 and 12 multiplied by 2. Therefore, the length of the hypotenuse is 13 multiplied by 2=26 centimeters.

Checking the answer:
a2+b2=c2
102+242=262
100+576=676
676=676 

Calculating triangles side lengths with Pythagorean theorem and triangles similarity

Similar triangles have the same shape, but not the same size (they have the same angle measures, but not the same side lengths).

The corresponding side lengths of similar triangles are related by a constant ratio, which is called k.

 Note that:
Similar triangles have the same angle measures.
If 2 lines are parallel, the triangles that are formed by them are similar.

Given parallel lines and a right triangle, we can calculate side lengths using triangles similarity combined with the Pythagorean theorem.

Calculating side ratio with triangles similarity

Consider the following example:

In the figure below the lines BE and CD are parallel. The angle measures and the side lengths (in centimeters) are given in the figure.

What are the lengths of the sides of triangle ACD?

Calculating right triangles side lengths

Calculating the length of AB:

Since BE is parallel to CD, the triangles ABE and ACD are similar. Therefore, the angle ACD is equal to the angle ABE and equal to 90 degrees.

Since the angle ABE is equal to 90 degrees, the triangle ABE is a right triangle. Therefore, we can apply the Pythagorean theorem in the triangle ABE:

The sides 3, 4 and 5 are Pythagorean triple, or we can solve the Pythagorean theorem:
a2+b2=c2
32+42=c2
c2=9+16
c2=25
c=5
AB=5 centimeters

Calculating the length of CD and DE:
AC=BC+AB
AC=5+5=10 centimeters

Since the lines BE and CD are parallel, the triangles ACD and ABE are similar. Therefore, the ratio of the related sides is k=AC/AB=10/5=2 (the side of the big triangle divided by the side of the small triangle is equal to 2).

Since k=2, AD/AE=2 and CD/BE=2.

Since AE=4 and k=2, AD/4=2, AD=8 centimeters and DE=AD-AE=8-4=4 centimeters.
Since BE=3 and k=2, CD/3=2, CD=6 centimeters.

Checking by calculating Pythagorean theorem in the big triangle ACD:
a2+b2=c2
62+82=102
This is a Pythagorean triple 3, 4 and 5 multiplied by 2.

Trigonometric ratios (sine, cosine and tangent)

Trigonometric ratios (functions) represent connections between angle degrees and side lengths in a right triangle.

The sine of an angle (sin) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, to the length of the hypotenuse. In the figure below in a right triangle ABC, sine (A)= BC/AC.

The cosine of an angle (cos) in a right triangle is defined as the ratio of the length of the side that is adjacent to the angle, to the length of the hypotenuse. In the figure below in a right triangle ABC, cosine (A)= AB/AC.

The tangent of an angle (tan) in a right triangle is defined as the ratio of the length of the side that is opposite to the angle, side that is adjacent to the angle. In the figure below in a right triangle ABC, tangent (A)= BC/AB.

In the figure below we see a right triangle ABC.
sin (A)=opposite/hypotenuse=BC/AC
cos (A)=adjacent/hypotenuse=AB/AC
tan (A)=opposite/adjacent=BC/AB

Sine and cosine of complementary angles

Complementary angles are two angles with the sum of 90 degrees.

Complementary angles in a right triangle: Since one angle in a right triangle is equal to 90 degrees and the sum of angles of a triangle is 180 degrees, the two other acute angles are complementary.

Sine of an angle in a right triangle is equal to cosine of its complementary angle. Meaning that given two complementary angles α and 90-α in a right triangle, sinα=cos(90-α).

Showing the connection between sine and cosine of complementary angles:
If we represent by α the measure of one angle in a right triangle, then the other angle measure is 90-α (complementary angles).

The figure below presents a right triangle ABC with complementary angles α and 90-α.

sinα=BC/AC

cos(90-α)=BC/AC

We see that sinα=cos(90-α)

Sine and cosine equality of complementary angles

Consider the following example:

What is sin (60) is equal to:
A. cos(30)
B. tan(60)
C. sin(30)
D. cos(60)

We know that sinα=cos(90-α), therefore, sin60=cos(90-60)=cos(30).
The answer A is correct.

Calculation of side lengths with sine, cosine and tangent

To calculate a side length, we need to know the value of the trigonometric function (sine, cosine or tangent) and the value of the other side.

Consider the following example:

The figure above represents a right triangle ABC.

If sin(a)= 0.5 and AC=5 centimeters, what is the value of AC?

sin(A)=BC/AC

0.5=BC/5

BC=0.5*5=2.5 centimeters

Calculation of sine, cosine and tangent in similar triangles

Similar triangles have the same angle measures and their corresponding side lengths are related by a constant ratio therefore they also have similar sine, cosine and tangent.

We can calculate sine, cosine or tangent in one triangle and conclude that they are identical in all similar triangles.

Related corresponding side lengths lead to similar sine, cosine and tangent:
If the sides of triangle 1 are x, y and z and the sides of a similar triangle 2 are kx,ky and kz (multiplied by a factor of k) then:
The sine of an angle A in triangle 1 is sin A1=x/z
The sine of an angle A in triangle 2 is sin A2=kx/kx=x/z
The sine of triangle 1 is equal to the sine of triangle 2.

Consider the following example:

In the figure below the lines BE and CD are parallel. The angle measures and the side lengths (in centimeters) are given in the figure.

What is the cosine of angle E?

Sine, cosine and tangent in similar triangles

Special right triangles

Special right triangles are right triangles whose sides are in a particular ratio. Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

The special right triangles with the side sizes length are given at the beginning of each SAT section.

30°, 60°, 90° triangle:
In a 30°, 60°, 90° right triangle the side opposite the 30° angle is half the length of the hypotenuse and the side opposite to 60° angle is equal to the length of a side opposite to 30° angle multiplied by √3.
In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

 45°, 45°, 90° triangle:
In a 45°, 45°, 90° right triangle the sides opposite the 45° angles are equal and the hypotenuse is equal to the side opposite to 45° angle multiplied by √2.
In 45°, 45°, 90° triangle the sides are s, s and s√2.

The following graphs present the special right triangles with the side sizes length.

Two special right triangles are 30°, 60°, 90° triangle and 45°, 45°, 90° triangle.

Calculation of side lengths given angle measures in special triangles

Given the length of any side in a special right triangle, we can calculate the length of the two other sides.

If you identify that the angles of a right triangle have the measures of 30,60,90 or 45,45,90 you know the ratios between their side length according to the graph presented above:

In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

In 45°, 45°, 90° triangle the sides are s, s and s√2.

 

Consider the following example:

In a right triangle the measures of the angles are 30°, 60°, 90° and the hypotenuse is equal to 8 centimeters.

What are the lengths of the other sides?

In 30°, 60°, 90° triangle the sides are x, x√3 and 2x.

Given the hypotenuse is equal to 8 centimeters, we know that the side opposite to 30° angle is equal to 8/2=4 centimeters. The side opposite to 60° angle is equal to 4*√3=√16*√3=√48.

Checking with Pythagorean theorem:
a2+b2=c2
42+√482=82
16+48=64
64=64

Consider the following example:

In a right triangle the measures of the angles are 45°, 45°, 90° and the hypotenuse is equal to √40 centimeters.

What are the lengths of the sides?

In 45°, 45°, 90° triangle the sides are s, s and s√2.

We are given that the hypotenuse is equal to √40 centimeters, therefore s√2=√40. Continue solving:
s√2=√40
s=√40/√2
s=√20

Checking with Pythagorean theorem:
a2+b2=c2
√202+√202=√402
20+20=40
40=40

Calculation of angle measures given side lengths in special triangles

Given the side lengths of a special right triangle, we can calculate the angle measures.

 If the ratio between a side and a hypotenuse in a right triangle is 1/2 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the side used in the ratio is located opposite to the 30° angle).

If the ratio between a side and a hypotenuse in a right triangle is √3/2 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the side used in the ratio is located opposite to the 60° angle).

If the ratio between two sides in a right triangle is √3 or 1/√3 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the bigger side which is multiplied by √3 is located opposite to the 60° angle).

If a right triangle is an isosceles triangle, it is a special right triangle with angles 45°, 45° and 90°.

If the ratio between a side and a hypotenuse in a right triangle is 1/√2 we know that the triangle is a special right triangle with 45°, 45°, 90° angles.

Consider the following example:

The ratio between two sides in a right triangle is √27/9, what are the measures of the angles of the triangle?

Simplifying the ratio √27/9 gives us:
√27/9=√27/√81
√27/√81=√27/(√27*√3)
√27/(√27*√3)=1/√3

If the ratio between two sides in a right triangle is 1/√3 we know that the triangle is a special right triangle with 30°, 60°, 90° angles (the bigger side which is multiplied by √3 is located opposite to the 60° angle).