Reference angle
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Reference Angle. Trigonometry MATH 103 S. Rook. Overview. Section 3.1 in the textbook: Reference angle Reference angle theorem Approximating with the calculator. Reference Angle. Reference Angle. One of the most important definitions in this class is the reference angle

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Reference angle

Reference Angle

Trigonometry

MATH 103

S. Rook


Overview

Overview

  • Section 3.1 in the textbook:

    • Reference angle

    • Reference angle theorem

    • Approximating with the calculator


Reference angle1

Reference Angle


Reference angle2

Reference Angle

  • One of the most important definitions in this class is the reference angle

    • Allows us to calculate ANY angle θusing an equivalent positive acute angle

      • We can now work in all four quadrants of the Cartesian Plane instead of just Quadrant I!

  • Reference angle: the positiveacute angle that lies between the terminal side of θ and the x-axis

    θ MUST be in standard position


Reference angle examples quadrant i

Reference Angle Examples – Quadrant I

Note that both θ and the reference angle are 60°


Reference angle examples quadrant ii

Reference Angle Examples – Quadrant II


Reference angle examples quadrant iii

Reference Angle Examples – Quadrant III


Reference angle examples quadrant iv

Reference Angle Examples – Quadrant IV


Reference angle summary

Reference Angle 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 θЄ QI:

        Ref angle = θ

      • If θ Є QII:

        Ref angle = 180° – θ

      • If θ Є QIII:

        Ref angle = θ – 180°

      • If θЄ QIV:

        Ref angle = 360° – θ


Reference angle summary continued

Reference Angle 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


Reference angle example

Reference Angle (Example)

Ex 1: Draw each angle in standard position and then name the reference angle:

a) 210°

b)101°

c)543°

d)-342°

e)-371°


Reference angle theorem

Reference Angle Theorem


Relationship between trigonometric functions with equivalent values

Relationship Between Trigonometric Functions with Equivalent Values

  • Consider the value of cos 60° and the value of cos 120°:

    cos 60° = ½ (Should have this MEMORIZED!)

    cos 120° = -½ (From Definition I with and

    30° – 60° – 90° triangle)


Relationship between trigonometric functions with equivalent values continued

Relationship Between Trigonometric Functions with Equivalent Values (Continued)

  • What is the reference angle of 120°?

    60°

  • Need to adjust the final answer depending on which quadrant θ terminates in:

    120° terminates in QII AND cos θ is negative in QII

  • Therefore, cos 120° = -cos 60° = -½

    • The VALUES are the same – just the signs are different!


Reference angle theorem1

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!


Reference angle summary1

Reference Angle Summary

  • Recall:

    • For any positive angle θ, 0° ≤ θ ≤ 360°

      • If θЄ QI:

        Ref angle = θ

      • If θ Є QII:

        Ref angle = 180° – θ

      • If θ Є QIII:

        Ref angle = θ – 180°

      • If θЄ QIV:

        Ref angle = 360° – θ


Reference angle summary continued1

Reference Angle Summary (Continued)

  • If θ > 360°:

    • Keep subtracting 360° from θ until 0° ≤ θ ≤ 360°

    • Go back to the first step

  • If θ < 0°:

    • Keep adding 360° to θ until 0° ≤ θ ≤ 360°

    • Go back to the the first step


Reference angle theorem example

Reference Angle Theorem (Example)

Ex 2: Use reference angles to find the exact value of the following:

a)cos 135°b) tan 315°

c)sec(-60°)d) cot 390°


Approximating with the calculator

Approximating with the Calculator


Approximating angles

Approximating Angles

  • Recall in Section 2.2 that we used sin-1, cos-1, and tan-1 to derive acute angles in the first quadrant

    • The Inverse Trigonometric Functions

  • Unlike the trigonometric functions, the Inverse Trigonometric Functions CANNOT be used to approximate every angle

    • We will see why when we cover the Inverse Trigonometric Functions in detail later


Approximating angles continued

Approximating Angles (Continued)

  • To circumvent this problem, we can use reference angles:

    • Find the reference angle that corresponds to the given value of a trigonometric function:

      • Recall that a reference angle is a positive acute angle which terminates in QI

      • Because cos θ and sin θ are both positive in QI, always use the POSITIVE value of the trigonometric function

    • Apply the reference angle by utilizing the quadrant in which θ terminates


Approximating angles example

Approximating Angles (Example)

Ex 3: Use a calculator to approximate θ if 0° < θ < 360° and:

a) cos θ = 0.0644, θЄ QIV

b) tan θ = 0.5890, θЄ QI

c) sec θ = -3.4159, θЄ QII

d) csc θ = -1.7876, θЄ QIII


Summary

Summary

  • After studying these slides, you should be able to:

    • Calculate the correct reference angle for any angle θ

    • Evaluate trigonometric functions using reference angles

    • Use a calculator and reference angles to approximate an angle θ given the quadrant in which it terminates

  • Additional Practice

    • See the list of suggested problems for 3.1

  • Next lesson

    • Radians and Degrees (Section 3.2)


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