A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor...

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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.

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:

4.4 Trig Functions of Any Angle2014

Objectives:Evaluate trigonometric functions of any angle. Use reference angles to evaluate trig functions.


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


6

Let (-3,4) be a point on the terminal side of . Find the sine, cosine and tangent of .

Precalculus

Example 1


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?


8

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


9

Given

Find the values of the six trig functions of

Precalculus

Example 2

tan 5

4 and cos 0


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

’

’

’


12Precalculus

Calculating Reference Angles

’

’

’

(degrees) 180'

(radians) '

(degrees) 180'

(radians) '

(degrees) 360'

(radians) 2'

Quadrant II Quadrant III Quadrant IV


13

Find the reference angle ’1) = 300°

2) =

3) = -135°

Precalculus

Example 4

37

12


15

Use reference angles to evaluate the trig function

1) 2)

3)

Precalculus

Example 4

cos43

tan 210

csc11

4


16

Let be an angle in Quadrant II such that sin =1/3. Find cos and tan .

Precalculus

Example 5 – Using Trig Identities

17

Explain how to use reference angles to determine trig functions of any angle

Precalculus4.4 Trigonometric Functions of Any Angle

Closure


18

4.4 pg 284 1-33 EOO, 53-73 odd, 91,93

Precalculus

Homework

A flagpole stands in the middle of a flat, level field. Fifty feet away from its base a surveyor...

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