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Copyright © 2011 Pearson, Inc.
4.2Trigonometric Functions of Acute Angles
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Slide 4.2 - 2 Copyright © 2011 Pearson, Inc.
What you’ll learn about
Right Triangle Trigonometry Two Famous Triangles Evaluating Trigonometric Functions with a
Calculator Applications of Right Triangle Trigonometry
… and whyThe many applications of right triangle trigonometry gave the subject its name.
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Slide 4.2 - 3 Copyright © 2011 Pearson, Inc.
Standard Position
An acute angle θ is in standard position, when one ray is along the positive x-axis, the vertex is at the origin, and the other ray extends into the first quadrant.
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Slide 4.2 - 4 Copyright © 2011 Pearson, Inc.
Trigonometric Functions Let θ be an acute angle in the right ΔABC. Then
sine θ( ) =sinθ =opphyp
cosecant θ( ) =cscθ =hypopp
cosine θ( ) =cosθ =adjhyp
secant θ( ) =secθ =hypadj
tangent θ( ) =tanθ =oppadj
cotangent θ( ) =cotθ =adjopp
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Slide 4.2 - 5 Copyright © 2011 Pearson, Inc.
Example Evaluating Trigonometric Functions of 45º
Find the values of all six trigonometric functions for an angle of 45º.
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Slide 4.2 - 6 Copyright © 2011 Pearson, Inc.
Example Evaluating Trigonometric Functions of 45º
Find the values of all six trigonometric functions for an angle of 45º.
sin 45o =opphyp
=12=
22
csc45o =hypopp
=21
cos45o =adjhyp
= 12=
22
sec45o =hypadj
=21
tan45o =oppadj
=11=1 cot45o =
adjopp
=11=1
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Slide 4.2 - 7 Copyright © 2011 Pearson, Inc.
Example Evaluating Trigonometric Functions of 60º
Find the values of all six trigonometric functions for an angle of 60º.
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Slide 4.2 - 8 Copyright © 2011 Pearson, Inc.
Example Evaluating Trigonometric Functions of 60º
Find the values of all six trigonometric functions for an angle of 60º.
sin60o =opphyp
=32
csc60o =hypopp
=23=2 33
cos60o =adjhyp
= 12 sec60o =
hypadj
=21=2
tan60o =oppadj
=31
cot60o =adjopp
=13=
33
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Slide 4.2 - 9 Copyright © 2011 Pearson, Inc.
Common Calculator Errors When Evaluating Trig Functions
Using the calculator in the wrong angle mode (degree/radians)
Using the inverse trig keys to evaluate cot, sec, and csc
Using function shorthand that the calculator does not recognize
Not closing parentheses
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Slide 4.2 - 10 Copyright © 2011 Pearson, Inc.
Example Solving a Right Triangle
A right triangle with a side length 6 includes a 27º angle
adjacent to the side of length 6. Find the measures of the
other two angles and the lengths of the other two sides.
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Slide 4.2 - 11 Copyright © 2011 Pearson, Inc.
Example Solving a Right Triangle
Since it is a right triangle, one of the other angles is 90o.
That leaves 180º −90º−27º=63º for the third angle.Use the labels on the figure to set up
equations to find a and b.
cos27º=6c
tan27º=b6
c=6
cos27ºb=6 tan27º
c≈6.73 b≈3.06
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Slide 4.2 - 12 Copyright © 2011 Pearson, Inc.
Quick Review
1. Solve for x.
x
3
2
2. Solve for x.
6
3
x
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Slide 4.2 - 13 Copyright © 2011 Pearson, Inc.
Quick Review
3. Convert 9.3 inches to feet.
4. Solve for a. 0.45 =a20
5. Solve for b. 1.72 =36b
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Slide 4.2 - 14 Copyright © 2011 Pearson, Inc.
Quick Review Solutions
1. Solve for x.
x
3
2
2. Solve for x.
6
3
x
13x =
3 3x =
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Slide 4.2 - 15 Copyright © 2011 Pearson, Inc.
Quick Review Solutions
3. Convert 9.3 inches to feet. 0.775 feet
4. Solve for a. 0.45 =a20
9
5. Solve for b. 1.72 =36b 900 / 43≈20.93