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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 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 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°

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 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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