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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.

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

4.4 Trigonometric Functions of Any Angle

6

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

Precalculus

Example 1

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?

4.4 Trigonometric Functions of Any Angle

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

4.4 Trigonometric Functions of Any Angle

9

Given

Find the values of the six trig functions of

Precalculus

Example 2

tan 5

4 and cos 0

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

’

’

’

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

4.4 Trigonometric Functions of Any Angle

13

Find the reference angle ’1) = 300°

2) =

3) = -135°

Precalculus

Example 4

37

12

4.4 Trigonometric Functions of Any Angle

15

Use reference angles to evaluate the trig function

1) 2)

3)

Precalculus

Example 4

cos43

tan 210

csc11

4

4.4 Trigonometric Functions of Any Angle

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

4.4 Trigonometric Functions of Any Angle

18

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

Precalculus

Homework