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September 29, 2014
T.4 Exact Values
Outcomes:
Demonstrateanunderstandingofanglesinstandardposition[0°to360°].
Solveproblems,usingthethreeprimarytrigonometricratiosforanglesfrom
0°to360°instandardposition.
WARM UP
If and is positive, then find the exact value of Untitled.notebook
September 29, 2014
y
­What is the measure of angle A?
1
­What is the length of the hypotenuse (as an exact value)?
A
x
1
­What are the values of sinA, cosA and tanA?
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September 29, 2014
y
­What is the height of the triangle?
2
60o
­1
2
60o
x
1
­Determine the values for sin60, cos60 and tan60.
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September 29, 2014
y
­What is the "length" of the triangle?
1
2
1
30o
x
1
­1
Determine the values of sin30, cos30 and tan30.
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September 29, 2014
How can we summarize this???
Special Triangles!!!
45o
60o
2
30o
2
1
1
30o
45o
1
60o
1
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September 29, 2014
Using Special Triangles...
Example 1
Write the following in terms of a reference angle and determine the exact value.
a) sin 210o
b) cos 300o
c) tan 225o
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What about for 0o and 90o ??
sin 0o = sin 90o = cos 0o =
cos 90o =
tan 0o =
tan 90o =
Why??
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Example 2
Determine the exact value(s) of A where:
b) tan A is undefined
a) cos A = ­ 2
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Example 3
The point P(­3, ) is on the terminal arm of an angle A.
a)DrawtheangleinstandardpositionandmarkthepointPontheterminalarm.
b)Statethevalueofx,yandr.
c)Determinetheexactvaluesofeachtrigratio.
d)Statethereferenceangleandtherotationangle.
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September 29, 2014
p. 174 #s 1, 2(pick 5), 3bc, 4
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September 29, 2014
45o
60o
2
30o
2
1
1
30o
45o
1
60o
1
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The Unit Circle!!!
September 29, 2014
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September 29, 2014
90o
120o
60o
135o
45o
150o
30o
0o 360o
180o
210o
330o
225o
315o
300o
240o
270o
Untitled.notebook
September 29, 2014
90o
120
o
60o
135
o
45o
150o
30o
0o 360o
180o
210o
330o
225o
315o
300o
240o
270o
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Some special notes about the unit circle...
September 29, 2014
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September 29, 2014
Using the unit circle...
Example 4
Use the unit circle to find the exact value of the following:
a) sin 240o
b) cos 240o
c) tan 240o
d) cos 135o
e) tan 120o
f) sin 180 o
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WARM UP
Determine the exact value of:
a) cos225
b) tan150
c) sin300
September 29, 2014
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Example 5
and are two points on the unit circle. If an object rotates counter­clockwise from point A to point B, through what angle has it rotated?
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Example 6
Determine the coordinates of the point on the unit circle that corresponds to a rotation angle of 148o.
Example 7
The point T(­0.8829, 0.4695) lies on the unit circle. Determine the value of A, where A is the angle made by the positive x­axis and the line passing through T.
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September 29, 2014
Special Triangles vs Unit Circle
Which is better???
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September 29, 2014
p. 174 #s 1, 2(pick 5), 3bc, 4
p. 176 #s 7 (pick 5), 8ac, 9bc, 10 OR 11
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Example 8
a) September 29, 2014
Use any method to find the exact values of:
b) c)