2.3 Evaluating Trigonometric Functions for any Angle JMerrill, 2009

Review from 2.2 Find the exact values of the other five trig functions for an angle θ in standard position, given 270 o 360 o θ

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

Positive, Negative or Zero? sin 240° Negative cos 300 o Positive tan 225 o 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 lies in Quadrant III, find sin and tan 2 θ -√3 θθθ

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. x y r

Ranges Continued Using a similar approach, we get:

Determining if a Value is Within the Range Evaluate (calculator) (not possible) (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 o = 135 o tan 495 o = tan 135 o. 135 o i s in Quad. II where tangent is negative. T he reference angle = 45 o 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 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 o and 321 o

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

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