Trigonometry MATH 103 S. Rook 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 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

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

Reference Angle Examples – Quadrant II

Reference Angle Examples – Quadrant III

Reference Angle Examples – Quadrant IV

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

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°

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

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)

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!

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 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

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, θ Є QIV b) tan θ = 0.5890, θ Є QI c) sec θ = -3.4159, θ Є QII d) csc θ = -1.7876, θ Є 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 Additional Practice See the list of suggested problems for 3.1 Next lesson Radians and Degrees (Section 3.2)