Reference Angle

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# Reference Angle - PowerPoint PPT Presentation

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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Presentation Transcript

### 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
• 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

Note that both θ and the reference angle are 60°

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)
• 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)

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

• 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)
• What is the reference angle of 120°?

60°

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 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 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 (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)

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 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)
• 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)

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
• 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