**Angles, arc length and trig functions topic includes 3 parts: **

Calculations of angles in radians.

Calculation of arc lengths and sector areas in radians.

Calculation of sine, cosine and tangent in radians.

**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.**