TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Section 4 4

TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Section 4. 4

OBJECTIVES: 1. Evaluate trigonometric functions of any angle. 2. Use reference angles to evaluate trigonometric functions.

Unit Circle Rationale 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 4 -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: 1. 213° 2. 1. 7 rad 3. − 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 any angle: 1. Find the trig value for the associated reference angle. 2. Pick the correct sign depending on where the terminal side lies. of

Exercise 5 Evaluate: 1. sin 5π/3 2. cos (− 60º) 3. 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. 575 -576 # 4 -84 (multiples of 4), # 87 -103 (odd)