4.2 & 4.4: Trig Functions and The Unit Circle Objectives: Identify a unit circle and describe its...
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Transcript of 4.2 & 4.4: Trig Functions and The Unit Circle Objectives: Identify a unit circle and describe its...
![Page 1: 4.2 & 4.4: Trig Functions and The Unit Circle Objectives: Identify a unit circle and describe its relationship to real #’s Evaluate trig functions using.](https://reader036.fdocuments.in/reader036/viewer/2022062304/56649ed95503460f94be7eb8/html5/thumbnails/1.jpg)
4.2 & 4.4: Trig Functions and The Unit Circle
Objectives:•Identify a unit circle and describe its relationship to real #’s•Evaluate trig functions using the unit circle•Use reference angles to evaluate trig functions for non-acute angles•Use domain and period to evaluate sine/cosine functions•Use a calculator to evaluate trig functions
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THE UNIT CIRCLE
• Circle with a radius of 1: x2 + y 2 = 1• Used to evaluate trig functions
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Each point on the unit circle (x,y) can also be used to find the 6 trig functions! This is huge!!
a. Draw a 60° angle in standard position.
b. Create a right triangle with the terminal side and the x-axis.
c. Find the other side lengths of the right triangle.
d. What is the sin (60°), cos (60°), tan (60°)?
e. What is the x coordinate on the unit circle? The y?
f. Notice anything???
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This also works for angles that are greater than 90⁰. To do this we use reference angles
Let Ө be an angle in standard position. Its reference angle is the acute angle, Ө’, formed by the
terminal side of Ө and the horizontal axis The trig function’s value for Ө is the same as the associated
reference angle, Ө’
TO FIND REFERENCE ANGLES:Quadrant 2: Quadrant 3: Quadrant 4:
180 180
360
2
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THE UNIT CIRCLE!!
Things to take notice of:x-coordinate is cos Ѳ, y-coordinate is sin ѲAn (x,y) ordered pair on the unit circle gives
you the sin and cos values, which will allow you to find other trig function values…AMAZING!!
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Activity
In small groups, find the sin, cos, and tan of the following angles (WITHOUT YOUR BOOKS!):
Draw central angle in standard position, radius = 1Create a special right triangle with the terminal side and the x-axis Calculate the sin Ѳ, cos Ѳ, and tan Ѳ.
6
11,
3
5,
3
4,
6
7,
6
5,
3
2,6
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Repeat with following angles:
4
7,
4
5,
4
3,4
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On your unit circle, label and their (x,y) coordinates. Which of the trig functions are undefined at these angles?
2,
2
3,,
2,0
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It’s Triggy
Getting Triggy With It!!
UNIT CIRCLE
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Fill in the Unit Circle
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Knowing the unit circle will help you tremendously. But you can always use special right triangles if you forget!
cos (-120°)
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Find the sin, cos, and tan for each real number, t.
1.
2.
3.
4.
5.
6.
2
t
3
t
4
5t
4
7t
180t
4
9t
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Definition of Trig Functions on The Unit Circle
t is a real number, (x,y) is the point on the unit circle corresponding to t:sin t = y csc t = 1/y , y ≠ 0cos t = x sec t = 1/x. x ≠ 0tan t = y/x, x≠0 cot t = x/y, y ≠ 0
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Determine the exact values of the 6 trig functions.
(-8/17, 15/17)
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DOMAIN and RANGE
Domain for sin and cos: All real numbers
Range:sin t = y cos t = x-1 < y < 1 -1 < x < 1
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The sin and cos function values repeat after . They are called periodic functions.
Definition of Periodic Functions:A function, f, is periodic if there exists a positive real number c such that
f(t + c) = f(t) (the value of the functions are the same)
for all t in the domain of f. The least number c for which f is periodic is called the period of f.
(Think about it…. and have the same sin and cos values)
2
6
2
6
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Examples: Evaluate
1. 2. 4
9sin
2
5cos
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EVEN and ODD Trig Functions
Remember, even functions: f(-x) = f(x) odd functions: f(-x) = - f(x)
cos and sec are evencos (-t) = cos t sec(-t) = sec t
sin, csc, tan, cot are oddsin(-t) = -sin t, csc(-t) = -csc t, tan (-t) = -tan t, cot(-t) = -cot (t)
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Examples
1. 2. Find sin (-t)= cos (-t)=
csc (-t)= sec (-t)=
3csc;3
1sin tt
5
1cos t
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Using what you know about the unit circle, why does it make sense that sin2θ + cos2 θ =1?