trigonometric ratios in the unit circle. warm-up (2 m) 1. sketch the following radian measures:
TRANSCRIPT
![Page 1: Trigonometric Ratios in the Unit Circle. Warm-up (2 m) 1. Sketch the following radian measures:](https://reader036.vdocuments.us/reader036/viewer/2022062407/56649cc15503460f949885b6/html5/thumbnails/1.jpg)
Trigonometric Ratios in the Unit Circle
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Warm-up (2 m)
1. Sketch the following radian measures:
6π17
65
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Trigonometric Ratios in the Unit Circle
The unit circle has a radius of 1
θtanxy
θtan
θcosrx
θcos
θsinry
θsin
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x is
y is
x is
y is
x is
y is
x is
y is
Quadrant IQuadrant II
Quadrant III Quadrant 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
5π
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Example: 18π31
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 #3:3π5
Trigonometric Ratio
Sine
Cosine
Tangent
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Example #4: 5π23
Trigonometric Ratio
Sine
Cosine
Tangent
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Your Turn:
Complete practice problems 4 – 7
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Reminder: Special Right Triangles
23
21 2
2
30°
60°
45°
45°
11
22
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 4π
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45° or 4π
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Summarizing Questions1. 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
43
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Example #6
65
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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
3π2
6π
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10.
Sine
Cosine
Tangent
11.
Sine
Cosine
Tangent
4π7
2π
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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
2
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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 #7:5π6
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Example #8:3π2
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Example #9:3π7
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Your Turn:
4π3
3π4
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Your Turn:
6π11
3π4
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Your Turn:
3π7
6π17
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Your Turn:
5π6
4π7
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Your Turn:
4π3
7π
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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 #10:6π7
Reference Angle:
Coterminal Angle:
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Example #10:Coordinates:
Sine:
Tangent:
Cosine:
6π7
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Example #11:
Reference Angle:
Coterminal Angle:3π7
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Example #11:Coordinates:
Sine:
Tangent:
Cosine:
3π7
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Example #12:
Reference Angle:
Coterminal Angle:3π17
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Example #12: 3π17
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:
1. 2. 3.3π7
sin6π7
cos
2π5
tan
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Summarizing QuestionsHow 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?
65
65
65
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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 ________.