Lesson 4.4 Trigonometric Functions of Any Angle Essential Question: How do you evaluate trigonometric functions of any angle?

Before we start… Find: sin 𝜃 cos 𝜃 How do you think you would find without a calculator: sin 60° cos 135°

Definition of Trigonometric Functions of Any Angle Let 𝜃 be an angle in standard position with 𝑥,𝑦 a point on the terminal side of 𝜃 and 𝑟= 𝑥 2 + 𝑦 2 ≠0. sin 𝜃= 𝑦 𝑟 cos 𝜃= 𝑥 𝑟 tan 𝜃= 𝑦 𝑥 , 𝑥≠0 cot 𝜃 = 𝑥 𝑦 , 𝑦≠0 sec 𝜃= 𝑟 𝑥 , 𝑥≠0 csc 𝜃 = 𝑟 𝑦 , 𝑦≠0

Let −3,4 be a point on the terminal side of 𝜃 Let −3,4 be a point on the terminal side of 𝜃. Find the sine, cosine, and tangent of 𝜃.

Let −2,3 be a point on the terminal side of 𝜃 Let −2,3 be a point on the terminal side of 𝜃. Find the sine, cosine, and tangent of 𝜃.

What about the signs of the trig functions? The signs of the trigonometric functions in the four quadrants can be determined from the definitions of the functions.

Coordinate Plane All Sin Tan Cos +

Given sin 𝜃=− 2 3 and tan 𝜃>0 , find cos 𝜃 and cot 𝜃 .

Given sin 𝜃= 4 5 and tan 𝜃<0 , find cos 𝜃 and csc 𝜃 .

Evaluate the sine and cosine functions at 0, 𝜋 2 , 𝜋, and 3𝜋 2 .

Evaluate the cosecant and cotangent functions at 0, 𝜋 2 , 𝜋, and 3𝜋 2 .

Definition of Reference Angle Let 𝜃 be an angle in standard position. Its reference angle is the acute angle 𝜃′ formed by the terminal side of 𝜃 and the horizontal axis.

Find the reference angle 𝜃′. 𝜃=213°

Find the reference angle 𝜃′. 𝜃=1.7

Find the reference angle 𝜃′. 𝜃=144°

Find the reference angle 𝜃′. 𝜃=300°

Find the reference angle 𝜃′. 𝜃=2.3

Find the reference angle 𝜃′. 𝜃=−135°

Reference Triangles You can use reference triangles to find the exact value of the trig functions. Reference triangles are quick easy relationships between the sides of the triangle with the special angles of 30˚, 60˚ and 45˚.

Reference Triangles

How do you find the trig function at any angle? Convert the angle to degrees if necessary. Draw the angle in the correct quadrant on the coordinate plane. Form a triangle (using the angle) perpendicular to the x-axis. Find the angle between the hypotenuse and the x-axis inside the drawn triangle. Label the triangle using a reference triangle. Choose the correct sides of the triangle for the needed ratio. Check the sign of the function in the quadrant. Reduce if possible. Don’t give decimal answers!

Evaluate. sin 5𝜋 3

Evaluate. cos −60°

Evaluate. tan 11𝜋 6

Evaluate. cos 4𝜋 3

Evaluate. tan −210°

Evaluate. csc 11𝜋 4

Evaluate. sin 135°

Evaluate. sec 2𝜋 3

Evaluate. cot 120°

Evaluate. cos 225°

Evaluate. tan 7𝜋 6

Let 𝜃 be an angle in Quadrant II such that sin 𝜃= 1 3 Let 𝜃 be an angle in Quadrant II such that sin 𝜃= 1 3 . Find (a) cos 𝜃 and (b) tan 𝜃 by using trigonometric identities.

Let 𝜃 be an angle in Quadrant III such that sin 𝜃=− 5 13 Let 𝜃 be an angle in Quadrant III such that sin 𝜃=− 5 13 . Find (a) sec 𝜃 and (b) tan 𝜃 by using trigonometric identities.

How do you evaluate trigonometric functions of any angle?

Ticket Out the Door Evaluate sec 5𝜋 3