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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:
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.

_________________        _________________
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:

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.

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).

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:

________________        _________________
2π                               360°

We can simplify the proportion getting:

_________________        _________________
π                                 180°

Consider the following example:

________________         ________________
π                                 180°

Represent radian measure by the variable x.

x    =    100
__         ____
π          180

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

Consider the following example:

________________         ________________
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.

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.

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.

Radian measures of angles of special right triangles

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

________________         ________________
π                                180°

x   =  30
__     ___
π      180

x=30π/180
x=π/6

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

________________         ________________
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.

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

________________         ________________
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.

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

________________         ________________
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).

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?

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).

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.

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.
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.

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.

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:
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.