4 4 Evaluating Trigonometric Functions for any Angle

4. 4 Evaluating Trigonometric Functions for any Angle Ref p. 288

Review Find the exact values of the other five trig functions for an angle θ in standard position, given 12 360 o θ 13 270 o -5

Positive Trig Function Values STUDENTS Sine and its reciprocal are positive ALL y -y r r -x y All functions are positive x r TAKE Tangent and its reciprocal are positive r -y CALCULUS Cosine and its reciprocal are positive

Positive, Negative or Zero? sin 240° cos 300 o tan 225 o Negative Positive

Determine the Quadrant In which quadrant is θ if cos θ and tan θ have the same sign? Quadrants I and II

Determine the Quadrant In which quadrant is θ if cos θ is negative and sin θ is positive? Quadrant II

Determine the Quadrant In which quadrant is θ if cot θ and sec θ have opposite signs? Quadrants III and IV

Using the Sign If and θ lies in Quadrant III, find sin θ and tan θ -1 θ -√ 3 2

Ranges of Trigonometric Functions We know that If the measure of increases toward 90 o, then y increases The value of y approaches r, and they are equal when So, y cannot be greater than r. Using the convenient point (0, 1) y can never be greater than 1. r y x

Ranges Continued Using a similar approach, we get:

Determining if a Value is Within the Range Evaluate (calculator) (not possible)

Reference Angles Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ ref angle Think of the reference angle as a “distance”—how close you are to the closest x-axis.

Find Reference Angle 150° 30° 225° 45° 300° 60°

Using Reference Angles a) sin 330° = = - sin 30° = - 1/2 b) cos 120° = = - cos 60° =-½

Using Reference Angles c) sin (-120°)= = - sin 60° d) Find the exact value of tan 495 o To find the correct quadrant, find the smallest positive coterminal angle. 495 o - 360 o = 135 o tan 495 o = tan 135 o is in Quad. II where tangent is negative. The reference angle = 45 o d) tan 495 o = - tan 45 o = -1

Finding Exact Measures of Angles Find all values of Sine is negative in QIII and QIV Using the 30 -60 -90 values we found earlier, we know

Finding Exact Measures of Angles – Cont. Our reference angle is 60 o. We must be 60 o off of the closest x-axis in QIII and QIV.

Approximating Approximate the value of 1. Ignore the negative and do 2. The answer is the reference angle, which we will round to 39 o 3. Sine is negative in QIII and QIV 4. 219 o and 321 o

Approximating Approximate the value of 1. 2. The answer is the reference angle, which we will round to 39 o 3. Sine is positive in QI and QII 4. 39 o and 141 o

You Do Find all values of Reference angle is 58 o 122 o and 238 o

4. 4 Trigonometric functions of any angle Review p. 294: 13 -23 odd, 29, 37