Trigonometric Functions of Any Angle

MATH 109 - PrecalculusS. Rook

Overview

• Section 4.4 in the textbook:– Trigonometric functions of any angle– Reference angles– Trigonometric functions of real numbers

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Trigonometric Functions of any Angle

Trigonometric Functions of Any Angle

• Given an angle θ in standard position and a point (x, y) on the terminal side of θ, then the six trigonometric functions of ANY ANGLE θ are can be defined in terms of x, y, and the length of the line connecting the origin and (x, y) denoted as r

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Trigonometric Functions of any Angle (Continued)

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Function Abbreviation Definition

The sine of θ sin θ

The cosine of θ cos θ

The tangent of θ tan θ

The cotangent of θ cot θ

The secant of θ sec θ

The cosecant of θ csc θ

Where and x and y retain their signs from (x, y)

r

y

r

x

0, xx

y

0, yy

x

0, xx

r

0, yy

r

22 yxr

Trigonometric Functions of any Angle (Continued)

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Function Abbreviation Definition

The sine of θ sin θ

The cosine of θ cos θ

The tangent of θ tan θ

The cotangent of θ cot θ

The secant of θ sec θ

The cosecant of θ csc θ

Where and x and y retain their signs from (x, y)

r

y

r

x

0, xx

y

0, yy

x

0, xx

r

0, yy

r

22 yxr

Algebraic Signs of Trigonometric Functions

• The sign of the six trigonometric functions depends on which quadrant θ terminates in:

r is the distance from the origin to (x, y) so it is ALWAYS positive

– The signs of x and y depend on which quadrant (x, y) lies

– Remember the shorthand notation involving “the element of” symbol:

• i.e. means theta is a standard angle which terminates in Q IV

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QIV

Algebraic Signs of Trigonometric Functions (Continued)

Functions θ Є QI θ Є QII θ Є QIII θ Є QIV

and + + – –

and + – – +

and + – + –

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r

ysin

y

rcsc

r

xcos

x

rsec

x

ytan

y

xcot

Trigonometric Functions of any Angle (Example)

Ex 1: Find the value of all six trigonometric functions if:

a) (-1, 2) lies on the terminal side of θ

b) (-7, -1) lies on the terminal side of θ

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Trigonometric Functions of any Angle (Example)

Ex 2: Given sec θ = -3⁄2 where cos θ < 0, find the exact value of tan θ and csc θ

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Reference Angles

Reference Angles

• An important definition is the reference angle– Allows us to calculate ANY angle θ using an

equivalent positive acute angle • We can now work in all four quadrants of the Cartesian

Plane instead of just Quadrant I!

• Reference angle: denoted θ’, the positive acute angle that lies between the terminal side of θ and the x-axis

θ MUST be in standard position

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Reference Angles Examples – Quadrant I

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Note that both θ and θ’ are 60°

Reference Angles Examples – Quadrant II

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Reference Angles Examples – Quadrant III

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Reference Angles Examples – Quadrant IV

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Reference Angles Summary

• Depending in which quadrant θ terminates, we can formulate a general rule for finding reference angles:– For any positive angle θ, 0° ≤ θ ≤ 360°:

• If θ Є QI:θ’ = θ

• If θ Є QII:θ‘ = 180° – θ

• If θ Є QIII:θ‘ = θ – 180°

• If θ Є QIV:θ’ = 360° – θ

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Reference Angles Summary (Continued)

– If θ > 360°:• Keep subtracting 360° from θ until 0° ≤ θ ≤ 360°• Go back to the first step on the previous slide

– If θ < 0°:• Keep adding 360° to θ until 0° ≤ θ ≤ 360°• Go back to the first step on the previous slide

– If θ is in radians:• Either replace 180° with π and 360° with 2π OR• Convert θ to degrees

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Reference Angles (Example)

Ex 3: i) draw θ in standard position ii) draw θ’, the reference angle of θ:

a) 312° b) π⁄8

c) 4π⁄5 d) -127°

e) 11π⁄3

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Trigonometric Functions of Real Numbers

Reference Angle Theorem

• Reference Angle Theorem: the value of a trigonometric function of an angle θ is EQUIVALENT to the VALUE of the trigonometric function of its reference angle– The ONLY thing that may be different is the sign

• Determine the sign based on the trigonometric function and which quadrant θ terminates in

– The Reference Angle Theorem is the reason why we need to memorize the exact values of 30°, 45°, and 60° only in Quadrant I!

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Evaluating a Trigonometric Function Exactly

• To evaluate a trigonometric function of θ:– Ensure that 0 < θ < 2π when using radians or

0° < θ < 360° when using degrees– Find θ’ the reference angle of θ– Evaluate the function using the EXACT values of

the reference angle and the quadrant in which θ terminates

• Write the function in terms of sine or cosine if necessary

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Evaluating a Trigonometric Function (Exactly)

Ex 4: Give the exact value:

a) sin 225° b) cos 750°

c) tan 120° d) sec -11π⁄4

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Summary

• After studying these slides, you should be able to:– Calculate the trigonometric function of ANY angle θ– State the reference angle of an angle θ in standard

position– Evaluate a trigonometric function using reference angles

and exact values• Additional Practice

– See the list of suggested problems for 4.4• Next lesson

– Graphs of Sine & Cosine Functions (Section 4.5)

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