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Section 9.8: Reference Angles 9.8: Reference Angles 9.8: Reference Angles • Associated with every angle drawn in the standard position (except quadrantal angles) there is another angle called the reference angle. angle • The reference angle is the positive acute angle formed by the terminal side of the given angle and the xx-axis. axis. Reference angles may appear in all four quadrants. Angles in quadrant I are their own reference angles. • Remember: Remember: The reference angle is measured from the terminal side to the xx-axis. (NOT THE Y-AXIS!) Y AXIS!) Chapter 9: Trig Part I 1 Section 9.8: Reference Angles Remember: Remember: The reference angle is measured from the terminal side to the x-axis. x axis. (NOT THE Y-AXIS!) Y AXIS!) Remember: Remember: The reference angle is measured from the terminal side to the x-axis. x axis. (NOT THE Y-AXIS!) Y AXIS!) Chapter 9: Trig Part I 2 Section 9.8: Reference Angles Remember: Remember: The reference angle is measured from the terminal side to the x-axis. x axis. (NOT THE Y-AXIS!) Y AXIS!) Remember: Remember: The reference angle is measured from the terminal side to the x-axis. x axis. (NOT THE Y-AXIS!) Y AXIS!) Chapter 9: Trig Part I 3 Section 9.8: Reference Angles Examples of Reference Angles Examples: Draw each angle in standard position and identify the reference angle. 1. 120° Chapter 9: Trig Part I 2. 330° 4 Section 9.8: Reference Angles Examples: Draw each angle in standard position and identify the reference angle. 3. 150° 4. −45° 5. Write cos 280º as a function of a positive acute angle. (Multiple Choice) a) cos 10° b) cos 80° c) cos 85° Chapter 9: Trig Part I 5 Section 9.8: Reference Angles 6. The value of cos 390º is equal to which of the following? a) sin 30° b) sin 60° c) -sin 30° 7. Write tan (-110º) as a function of a positive acute angle. (Multiple Choice) a) tan 10° b) tan 20° c) tan 70° Chapter 9: Trig Part I 6 Section 9.8: Reference Angles More Examples of Reference Angles 8. For each indicated value and quadrant, find the angle measure to the nearest degree, on the interval [0,360). (Always use the positive ratio measure when putting into calculator.) tanθ = −4.7047 Quadrant II 102° cos θ = 0.9816 Quadrant IV 349° sinθ = −0.2756 Quadrant IV 344° Chapter 9: Trig Part I 7 Section 9.8: Reference Angles Finding Exact Values of Trigonometric Functions 2. Find the exact value of each expression: sin 300° Step 1: Sketch the angle and determine the measure of the reference angle. sin 60° Finding Exact Values of Trigonometric Functions 2. Find the exact value of each expression: sin 300° Step 2: Determine the sign of the function (positive or negative). sin 300° − sin 60° Chapter 9: Trig Part I 8 Section 9.8: Reference Angles Finding Exact Values of Trigonometric Functions 2. Find the exact value of each expression: sin 300° Step 3: Evaluate the function. [Use special right triangles or the unit circle] − sin 60° 3 =− 2 3. Find the exact value of each expression: cos 135° − cos 45° =− Chapter 9: Trig Part I 2 2 9 Section 9.8: Reference Angles 4. Find the exact value of each expression: tan 240 ° + tan 60° = 3 5. Find the exact value of each expression: sin( −135)° − sin 45° =− Chapter 9: Trig Part I 2 2 10 Section 9.8: Reference Angles 6. Find the exact value of each expression: cos(−210°) − cos 30° =− 3 2 7. Find the exact value of each expression: tan 330° − tan 30° =− Chapter 9: Trig Part I 3 3 11