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Reference Angle

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

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  1. 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 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

  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 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

  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° • Go back to the first step on the previous slide

  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 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

  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 = 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

  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)

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