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Finding the Exact Value of Trigonometric Functions
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Transcript of Finding the Exact Value of Trigonometric Functions
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Finding the Exact Value of Trigonometric Functions
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Review: Special Right Triangles
160°
30°
1
2
3
2
1
45°
45°
2
2
2
2
π / 6
π / 3
π / 4
π / 4
Find the missing side Lengths:
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0
3π / 2
π / 2
Important Points on Unit Circle
-1
-1
1
1 0,1
1,0 1,0
30°
45°
60°
150°
135°
120°
210°
225°
240°
330°
315°
300°
0°180°
90°
π / 6
270°
π / 4π / 3
0, 1
2π / 33π / 4
5π / 6
π
7π / 6
5π / 4
4π / 3 5π / 37π / 4
11π / 6
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1
45°2
2
2
2 1/21
30°3
21/2
1
60°
3
2
Important Points on Unit Circle
-1
-1
1
1
3 1,
2 2
1 3,
2 2
2 2,
2 2
0,1
3 1,
2 2
1 3,
2 2
2 2
,2 2
1,0
3 1,
2 2
1 3,
2 2
2 2,
2 2
0, 1
3 1,
2 2
1 3,
2 2
2 2,
2 2
1,0
π / 6
π / 4π / 3
π / 22π / 3
3π / 4
5π / 6
7π / 6
5π / 4
4π / 33π / 2
5π / 37π / 4
11π / 6
0π
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Important Points on Unit Circle
-1
-1
1
1
3 1,
2 2
1 3,
2 2
2 2,
2 2
0,1
3 1,
2 2
1 3,
2 2
2 2
,2 2
1,0
3 1,
2 2
1 3,
2 2
2 2,
2 2
0, 1
3 1,
2 2
1 3,
2 2
2 2,
2 2
1,0
30°
45°
60°
150°
135°
120°
210°
225°
240°
330°
315°
300°
0°180°
90°
270°
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Important Points on Unit Circle
-1
-1
1
1
3 1,
2 2
1 3,
2 2
2 2,
2 2
0,1
3 1,
2 2
1 3,
2 2
2 2
,2 2
1,0
3 1,
2 2
1 3,
2 2
2 2,
2 2
0, 1
3 1,
2 2
1 3,
2 2
2 2,
2 2
1,0
π / 6
π / 4π / 3
π / 22π / 3
3π / 4
5π / 6
7π / 6
5π / 4
4π / 33π / 2
5π / 37π / 4
11π / 6
0π
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Reference AngleOn the left are 3 reference angles that we know exact trig values
for. To find the reference angle for angles not in the 1st quadrant (the angles at right), ignore the integer in the numerator.
0:6
3
5:4
4
0:3
6
5 7 11, ,
6 6 6
3 5 7, ,
4 4 4
2 4 5, ,
3 3 3
Then multiply the number in
the numerator
by the degree to find the angle’s
quadrant.
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Stewart’s Table: Finding Exact Values of Trig Functions
R.A. Sin Cos Tan
0
6
4
3
2
0
2
1
2
2
23
2
4
2
13
22
21
2
0
1. Find the value of the Reference Angle.
2. Find the angles quadrant to figure out the sign (+/-).
0
1
2
2
21
Each time the square root number goes up by 1
Reverse the order of the values from sine
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Example 1
Find the exact value of the following:
34cos
Reference Angle:
Cosine of Reference Angle:
3 45 135
Sign of Cosine in Second Quadrant:
Second Quadrant
4
4cos
Quadrant of Reference Angle:
22
, ,
, ,
Negative
Therefore: 234 2cos
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Example 2
Solve: 2sin 1 0x
1 12 2Reference Angle that Makes sin = = True:x
76 6 Solutions in the interval 0 2 :x
6
116 62
All solutions:76
116
2sin 1x 1
2sin x
2 where is an integer
2
nn
n
π / 6π / 6
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-1
Slope on the Unit Circle
-1 1
1
Ө
(cosӨ,sinӨ)
cosӨ
sinӨ
Slope =
sin
cos
tan
Opposite
Adjacent
What is the slope of the terminal side of an angle on the unit circle?
opposite
adjacent
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A Definition of Tangent
There are values for which the tangent function are undefined:
sintan
cos
2 5
2,
2 n For any integer n.
32, 9
2, 72, ,...11
2,
The tangent function is defined as:
Any Θ that makes cos(Θ)=0.
In general:
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Stewart’s Table: Finding Exact Values of Trig Functions
R.A. Sin Cos Tan
0
6
4
3
2
00
2
1 1
2 2
2
23
2
4 21
2 2
13
22
21
2
0
0
1
1 2
3 2
1
0
2 2
2 2
3 2
1 2
1. Find the value of the Reference Angle.
2. Find the angles quadrant to figure out +/-.
0
1 2
2 3
1
1
3 3
3
3 2
2 1 3
Each time the square root number goes up by 1
Reverse the order of the values from sine
sintan
cos
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Example 3
Find the exact value of the following:
53tan
Reference Angle:
Tangent of Reference Angle:
5 60 300
Sign of Tangent in Fourth Quadrant:
Fourth Quadrant
3
3tan
Quadrant of Angle:
3
, ,
, ,
negativepositive Negative
Therefore: 53tan 3
Tho
ught
pro
cess
The only thing required for a correct answer (unless the question says explain)