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![Page 1: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/1.jpg)
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2
Example: In traveling across flat land you notice a mountain directly in front of you. Its angle of elevation to the peak is 3.5 degrees. After you drive 13 miles closer to the mountain, the angle of elevation is 9 degrees.Approximate the height of the mountain.
![Page 2: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/2.jpg)
4.4 Trigonometric Functions of Any Angle
3
At a certain distance, the angle of elevation to the top of a tree is 60 degrees. From 40 feet further back the angle of elevation is 45 degrees. Find the height of the building.
To receive credit you must have a diagram completely labeled, the trig function that you used to solve, and the answer.
Precalculus
Ex:
![Page 3: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/3.jpg)
4.4 Trig Functions of Any Angle2014
Objectives:Evaluate trigonometric functions of any angle. Use reference angles to evaluate trig functions.
![Page 4: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/4.jpg)
4.4 Trigonometric Functions of Any Angle
5
Given an angle in standard position with (x,y) a point on the terminal side of
Precalculus
Trig Functions of Any Angle
sin y
r
cos x
r
tan y
x, x 0
csc r
y, y 0
sec r
x, x 0
cot x
y, y 0
r
(x,y)
y
x
r x 2 y 2
![Page 5: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/5.jpg)
4.4 Trigonometric Functions of Any Angle
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Let (-3,4) be a point on the terminal side of . Find the sine, cosine and tangent of .
Precalculus
Example 1
![Page 6: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/6.jpg)
4.4 Trigonometric Functions of Any Angle
7
One way to think of the sign of a function is to remember the variable it is defined by. Since the radius is always positive, only the signs of x and y influence the sign of the function.
In any given quadrant:
cosine and secant have the same sign as x
sine and cosecant have the same sign as y
tangent and cotangent have the same sign as the ratio of x and y
Precalculus
How do I know the sign of the trig function?
![Page 7: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/7.jpg)
4.4 Trigonometric Functions of Any Angle
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1)
2)
3)
4)
Precalculus
Determine the Quadrant in Which the Angle Lies
cos 0and sin < 0
tan 0and sin < 0
sec 0and cot < 0
tan 0and csc < 0
![Page 8: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/8.jpg)
4.4 Trigonometric Functions of Any Angle
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Given
Find the values of the six trig functions of
Precalculus
Example 2
tan 5
4 and cos 0
![Page 9: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/9.jpg)
4.4 Trigonometric Functions of Any Angle
11
Let be an angle in standard position. Its reference angle is the acute angle ’ formed by the terminal side of and the x-axis.
Precalculus
Reference Angles
’
’
’
![Page 10: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/10.jpg)
4.4 Trigonometric Functions of Any Angle
12Precalculus
Calculating Reference Angles
’
’
’
(degrees) 180'
(radians) '
(degrees) 180'
(radians) '
(degrees) 360'
(radians) 2'
Quadrant II Quadrant III Quadrant IV
![Page 11: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/11.jpg)
4.4 Trigonometric Functions of Any Angle
13
Find the reference angle ’1) = 300°
2) =
3) = -135°
Precalculus
Example 4
37
12
![Page 12: A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor measures the angle to the top of the flagpole as 48°.](https://reader036.fdocuments.in/reader036/viewer/2022081816/5697bfa71a28abf838c9900a/html5/thumbnails/12.jpg)
4.4 Trigonometric Functions of Any Angle
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Use reference angles to evaluate the trig function
1) 2)
3)
Precalculus
Example 4
cos43
tan 210
csc11
4
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4.4 Trigonometric Functions of Any Angle
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Let be an angle in Quadrant II such that sin =1/3. Find cos and tan .
Precalculus
Example 5 – Using Trig Identities
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Explain how to use reference angles to determine trig functions of any angle
Precalculus4.4 Trigonometric Functions of Any Angle
Closure
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4.4 Trigonometric Functions of Any Angle
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4.4 pg 284 1-33 EOO, 53-73 odd, 91,93
Precalculus
Homework