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Trigonometric Ratios in the Unit Circle

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Warm-up (2 m) 1. Sketch the following radian measures:

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Trigonometric Ratios in the Unit Circle The unit circle has a radius of 1

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x is y is x is y is x is y is x is y is Quadrant IQuadrant II Quadrant IIIQuadrant IV

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“All Students Take Calculus” AS CT all ratios are positive sine is positive tangent is positive cosine is positive cosecant is positive cotangent is positive secant is positive

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Example: Trigonometric Ratio Sine Cosine Tangent

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Example: Trigonometric Ratio Sine Cosine Tangent

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Your Turn: Complete problems 1 - 3

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Sketching Negative Radians and/or Multiple Revolutions 1. Whenever the angle is less than 0 or more than 2 pi, solve for the coterminal angle between 0 and 2 pi 2. Sketch the coterminal angle

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Example #3: Trigonometric Ratio Sine Cosine Tangent

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Example #4: Trigonometric Ratio Sine Cosine Tangent

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Your Turn: Complete practice problems 4 – 7

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Reminder: Special Right Triangles 30° 60° 45° 30° – 60° – 90°45° – 45° – 90°

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Investigation! Fit the paper triangles onto the picture below. The side with the * must be on the x-axis. Use the paper triangles to determine the coordinates of the three points.

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Special Right Triangles & the Unit Circle

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Special Right Triangles & the Unit Circle: 30°- 60°

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30°- 60°

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45° or

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Summarizing Questions 1. In which quadrants is tangent positive? Why? 2. In which quadrants is cosecant negative? Why? 3. How do I sketch negative angles? 4. How can I sketch angles with multiple revolutions? 5. What are some ways of remembering the radian measures of the Unit Circle? 6. How do we get the coordinates for π/6, π/4, and π/3?

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Example #5

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Example #6

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Your Turn: Use your unit circle to solve for the exact values of sine, cosine, and tangent of problems 8 – 11. Rationalize the denominator if necessary.

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8. Sine Cosine Tangent 9. Sine Cosine Tangent

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10. Sine Cosine Tangent 11. Sine Cosine Tangent

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Reference Angles Reference angles make it easier to find exact values of trig functions in the unit circle Measure an angle’s distance from the x-axis

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Reference Angles, cont. Always Coterminal Acute (less than ) Have one side on the x-axis

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Solving for Reference Angles Step 1: Calculate the coterminal angle if necessary (Remember, coterminal angles are positive and less than 2π.) Step 2: Sketch either the given angle (if less than 2π) or the coterminal angle (if greater than 2π) Step 3: Determine the angle’s distance from the x-axis (It is almost always pi/denominator!!!) This is the reference angle!!!!

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Example #7:

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Example #8:

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Example #9:

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Your Turn:

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Solving for Exact Trig Values Step 1: Solve for the coterminal angle between 0 and 2π if necessary Step 2: Solve for the reference angle (Note the quadrant) Step 3: Identify the correct coordinates of the angle (Make sure the signs of the coordinates match the quadrant!) Step 4: Solve for the correct trig ratio (Rationalize the denominator if necessary)

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Example #10: Reference Angle: Coterminal Angle:

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Example #10: Coordinates: Sine: Tangent: Cosine:

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Example #11: Reference Angle: Coterminal Angle:

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Example #11: Coordinates: Sine: Tangent: Cosine:

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Example #12: Reference Angle: Coterminal Angle:

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Example #12: Coordinates: Sine: Tangent: Cosine:

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Your Turn: Complete problems 12 – 18.

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Exit Ticket Solve for the exact values of the following:

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Summarizing Questions How do we get the coordinates for using the 45° – 45° – 90° triangle? Why are the coordinates of negative? What are the sine, cosine, and tangent of ? What is a reference angle?

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Exit Ticket – “The Important Thing” On a sheet of paper (with your name!) complete the sentence below: Three important ideas/things from today’s lesson are ________, ________, and ________, but the most important thing I learned today was ________.

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