2.3 Evaluating Trigonometric Functions for any Angle

2.3 Evaluating Trigonometric Functions for any Angle

Evaluating Trigonometric Functions for any Angle Ref p. 288 Review Find the exact values of the other five trig functions for an angle in standard position, given o 5 sin , 270 360o 13 1 2

13 270o 13 csc 5 12 cos 13 13 sec 12

5 tan 12 12 cot 5 360o -5 Positive Trig Function Values STUDENTS Sine and its

reciprocal are y positive ALL r r -x y All functions

are positive x -y r TAKE Tangent and its reciprocal are positive r -y

CALCULUS Cosine and its reciprocal are positive Positive, Negative or Zero? sin 240 cos 300o tan 225o Negative

Positive Positive Determine the Quadrant In which quadrant is if cos and tan have the same sign? Quadrants I and II Determine the Quadrant In which quadrant is if cos is negative and sin is positive? Quadrant II

Determine the Quadrant In which quadrant is if cot and sec have opposite signs? Quadrants III and IV Using the Sign 1 cos and lies in Quadrant III, find sin 2 If 3

sin 2 -1 -3 2 tan 3 and tan Ranges of

Trigonometric Functions y We know that sin r If the measure of increases toward 90o, then y increases 90 The value of y approaches r, o 90 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. o r y x Ranges Continued Using a similar approach, we get:

1 sin 1 1cos 1 sec 1 or sec 1 csc 1 or csc 1 tan and cot can be any real number Determining if a Value is Within the Range Evaluate (calculator) cos 2 cot 0 3 sin 2

(not possible) 90o (not possible) Reference Angles Reference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of ref angle ref angle ref angle

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 =- Using Reference Angles c) sin (-120)= = - sin 60 3 2

d) Find the exact value of tan 495o To find the correct quadrant, find the smallest positive coterminal angle. 495o - 360o = 135o tan 495o = tan 135o. 135o is in Quad. II where tangent is negative. The reference angle = 45o tan 495o = - tan 45o = -1 Finding Exact Measures of Angles Find all values of 3 , where 0 360 , when sin 2

o o sin 60o 3 2 Sine is negative in QIII and QIV Using the 30-60-90 values we found earlier, we know Finding Exact Measures of Angles Cont.

3 sin 60 2 o Our reference angle is 60o. We must be 60o off of the closest x-axis in QIII and QIV. o 240 and 300 o Approximating Approximate the value of

, if sin .6293 1. Ignore the negative and do sin (.6293) 2. The answer is the reference angle, which we 38.99849667 will round to 39o 1 3. Sine is negative in QIII and QIV 4. 219o and 321o

Approximating Approximate the value of 1. , if sin .6293 sin 1 (.6293) 38.99849667 2. The answer is the reference angle, which we will round to 39o 3. Sine is positive in QI and QII 4. 39o and 141o You Do

Find all values of o o , where 0 360 when cos 0.5299 Reference angle is 58o 122o and 238o 4.4 Trigonometric functions of any angle Review p.294: 13-23 odd, 29, 37

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