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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 2 Reference Angle 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 4 Reference Angle Examples – Quadrant I Note that both θ and the reference angle are 60° 5 Reference Angle Examples – Quadrant II 6 Reference Angle Examples – Quadrant III 7 Reference Angle Examples – Quadrant IV 8 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 9 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 10 Reference Angle (Example) Ex 1: Draw each angle in standard position and then name the reference angle: a) b) c) d) e) 210° 101° 543° -342° -371° 11 Reference Angle Theorem 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 1, 3 and 30° – 60° – 90° triangle) 13 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! 14 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! 15 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 16 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 17 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° 18 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 20 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 21 Approximating Angles (Example) Ex 3: Use a calculator to approximate θ if 0° < θ < 360° and: a) b) c) d) cos θ = 0.0644, θ Є QIV tan θ = 0.5890, θ Є QI sec θ = -3.4159, θ Є QII csc θ = -1.7876, θ Є QIII 22 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) 23