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Chapter 4 Trigonometry

Section 4 Trigonometric

Functions of Any Angle

to infinity and beyond . . . !

Math Analysis

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Objectives Evaluate trigonometric functions of any

angle to ± infinity

Use reference angles to evaluate trigonometric functions.

Use trig models to solve real world problems

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Definitions of trigonometric functions

Syr Cxr Tyx

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Find the trig functions using a point on the terminal side syr cxr tyx

• (-3, 4)

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Quadrant angles

θ cos θ sin θ

0

π/2

π

3π/2

2π

x y

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Trigonometric function signs

AllStar

Trig Class

Sin/csc

Tan/cot Cos/sec

All 6 trig functions are positive in quadrant 1

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Find the quadrant1. sinθ > 0; cosθ <0

2. sinθ < 0; cosθ > 0

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Reference angleLet θ be an angle in standard position. Its reference angle is the acute angle θ‘ formed by the terminal side of θ and the horizontal axis.

Reference angles are always measured between the x axis and the terminal side of the angle (always +!)

Notice the butterfly shape

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Drawing Reference AnglesFind the reference angle θ', and sketch θ and θ' in standard position.

1) θ = -145° 2)11

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Finding the Trig functions of any angle1. Find the reference angle

2. Determine the value of the given trig function of the reference angle.

3. Determine the sign—based on the quadrant of the given (original) angle

tan135

180 135

45

ref

tan 45 1

135 II Q

tan IIis in Q

tan135 1

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Now: Find all 6 trig functions of an angle

Find the six trigonometric functions of θ with the given constraint.

15tan ; sin 0

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Now, let’s make it tougher

The terminal side of θ lies on the line y = (1/3)x in quadrant I. Find the values of the 6 trig functions of θ by finding a point on the line.

sin θ =

cos θ =

csc θ =

sec θ =

tan θ =

cot θ =

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Try These:

Find the values of the six trig functions of θ with the given constraints.

sin θ = 0; where sec θ = -1

Find the reference angle θ‘ and sketch it and the angle θ in standard position. a) θ = 3.5 b) θ = 750°

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Find the trig functions of an angle

Evaluate the sine, cosine, and tangent of the angle without using a calculator.

-405°

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Do 88(a)&(c)

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Do 91

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Objectives Evaluate trigonometric functions of any angle to

± infinity Syr Cxr Tyx Quadrantal Angles

Use reference angles to evaluate trigonometric functions.

1. Find ref <2. Find trig value of ref <3. Determine sign of trig value for given <

Use trig models to solve real world problems Draw, identify, solve

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Pg. 284: 3-52 (multiples of 3-- i.e. 3, 6, 9, etc.)