TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Section 4.4

Objectives: Evaluate trigonometric functions of any angle. Use reference angles to evaluate trigonometric functions.

Unit Circle Rationale What if the radius (hypotenuse) is not 1? Recall: when using the unit circle to evaluate the value of a trigonometric function, cos θ = x and sin θ = y. Actually, since the radius (hypotenuse) is 1, the trigonometric values are really cos θ = x/1 and sin θ = y/1. What if the radius (hypotenuse) is not 1?

Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.

Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ. The triangle formed by the point (4, 3) is similar to a smaller triangle in the unit circle. To get to that unit circle triangle, we need to scale down the larger triangle by dividing by the scale factor 5 (the length of the larger hypotenuse).

Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ. Solution:

Trigonometric Functions of Any Angle Let (x, y) be a point on the terminal side of an angle θ in standard position with

Exercise 2 Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trigonometric functions for θ.

Signs of Trigonometric Functions The sign of every trig. function depends on a quadrant (x, y) lies within. (r is always positive!!!!)

Exercise 3 Given sin θ = 4/5 and tan θ < 0, find cos θ and csc θ. 5 -3

Reference Angles When building the unit circle, for 120º we drew a triangle with the x-axis to form a 60º angle. This 60º angle was the reference angle for 120º .

Reference Angles Let θ be an angle in standard position. It’s reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis.

Reference Angles Let θ be an angle in standard position. It’s reference angle is the positive acute angle θ’ formed by the terminal side of θ and the x-axis.

Exercise 4 Find the reference angle for each of the following: 213° 1.7 rad −144°

Reference Angles When an angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0º and 360º (or 0 and 2π).

Reference Angles How Reference Angles Work: Same except maybe a difference of sign.

Reference Angles To find the value of a trigonometric function of any angle: Find the trig value for the associated reference angle. Pick the correct sign depending on where the terminal side lies.

Exercise 5 Evaluate: sin 5π/3 cos (−60º) tan 11π/6

Exercise 6 Let θ be an angle in Quadrant III such that sin θ = −5/13. Find: a) sec θ and b) tan θ using trigonometric identities.

Homework Read Section 4.4, Complete pg. 499-500 # 4-84 (multiples of 4), # 88-92 (e), 93-103 (odd)