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Reference Angle

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Reference Angle

Trigonometry

MATH 103

S. Rook

- Section 3.1 in the textbook:
- Reference angle
- Reference angle theorem
- Approximating with the calculator

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!

- Allows us to calculate ANY angle θusing an equivalent positive acute angle
- Reference angle: the positiveacute angle that lies between the terminal side of θ and the x-axis
θ MUST be in standard position

Note that both θ and the reference angle are 60°

- 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

- If θЄ QI:

- For any positive angle θ, 0° ≤ θ ≤ 360°:

- 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

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)

- 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 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!

- The ONLY thing that may be different is the sign

- 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

- If θЄ QI:

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

- 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

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

- 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

- 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

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

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

- 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)