1. Trigonometry MATH 103 S. Rook
Reference Angle
Trigonometry
MATH 103
S. Rook

2. Overview Section 3.1 in the textbook: Reference angle
Reference angle theorem
Approximating with the calculator

3. Reference Angle

4. Reference Angle
One of the most important definitions in this class 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: the positive acute angle that lies between the terminal side of θ and the x-axis
θ MUST be in standard position

5. Reference Angle Examples – Quadrant I
Note that both θ and the reference angle are 60°

6. Reference Angle Examples – Quadrant II

7. Reference Angle Examples – Quadrant III

8. Reference Angle Examples – Quadrant IV

9. 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 θ Є QI:
Ref angle = θ
If θ Є QII:
Ref angle = 180° – θ
If θ Є QIII:
Ref angle = θ – 180°
If θ Є QIV:
Ref angle = 360° – θ

10. 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°

11. 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°

12. Reference Angle Theorem

13. 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)

14. 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!

15. 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!

16. Reference Angle Summary
Recall:
For any positive angle θ, 0° ≤ θ ≤ 360°
If θ Є QI:
Ref angle = θ
If θ Є QII:
Ref angle = 180° – θ
If θ Є QIII:
Ref angle = θ – 180°
If θ Є QIV:
Ref angle = 360° – θ

17. 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

18. 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°

19. Approximating with the Calculator

20. 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

21. 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

22. Approximating Angles (Example)
Ex 3: Use a calculator to approximate θ if 0° < θ < 360° and: a) cos θ = , θ Є QIV b) tan θ = , θ Є QI c) sec θ = , θ Є QII d) csc θ = , θ Є QIII

23. 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)