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7.4 Trigonometric Functions of General Angles

In this section, we will study the following topics:

Evaluating trig functions of any angle

Using the unit circle to evaluate the trig functions of quadrantal angles

Finding coterminal angles

Using reference angles to evaluate trig functions.

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In 7.3, we looked at the definitions of the trig functions of acute angles

of a right triangle. In this section, we will expand upon those definitions

to include ANY angle.

We will be studying angles that are greater than 90° and less than 0°,

so we will need to consider the signs of the trig functions in each of the

quadrants.

We will start by looking at the definitions of the trig functions of any

angle.

Trig Functions of Any Angle

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Definitions of Trigonometric Functions of Any Angle

Let be an angle in standard position with (x, y) a point on the

terminal side of and

Definitions of Trig Functions of Any Angle

2 2r x y

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

y

x

(x, y)

r

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Let (-12, -5) be a point on the terminal side of . Find the exact values of the six trig functions of .

Example*

r-5

y

x

(-12, -5)

-12

First you must find the value of r:

2 2r x y

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Example (cont)

13-5

y

x

(-12, -5)

-12

sin

cos

tan

csc

sec

cot

y

rx

ry

xr

y

r

xx

y

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Let (-3, 7) be a point on the terminal side of . Find the value of the six trig functions of .

You Try!

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Since the radius is always positive (r > 0), the signs of

the trig functions are dependent upon the signs of x

and y.

Therefore, we can determine the sign of the functions

by knowing the quadrant in which the terminal side of

the angle lies.

The Signs of the Trig Functions

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The Signs of the Trig Functions

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A trick to remember where each trig function is POSITIVE:

A

CT

S

All Students Take Calculus

Translation:

A = All 3 functions are positive in Quad 1

S= Sine function is positive in Quad 2

T= Tangent function is positive in Quad 3

C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tan is positive, but sine and cosine are negative; ...

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is positive, secant is positive wherever cosine is positive, …

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Determine if the following functions are positive or negative:

Example

sin 210°

cos 320°

cot (-135°)

csc 500°

tan 315°

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Given and , find the values of the five other

trig function of .

Example*

8cos

17 cot 0

8cos

17

x

r

2 2 2

2 22

Using the fact that , we can find .

-8 17

x y r y

y

Solution

First, determine the quadrant in which lies. Since the cosine is negative and the cotangent is positive, we know that lies in Quadrant _____ .

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Now we can find the values of the remaining trig functions:

Example* (cont)

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

8 15 17x y r

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Given and , find the values of the five

other trig functions of .

Another Example

3cot

8 2

16

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Trig functions of Quadrantal Angles

To find the sine, cosine, tangent, etc. of angles whose terminal side

falls on one of the axes , we will use the unit

circle.

3(..., , , 0, , , , 2 ,...)

2 2 2

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

2

3

2

Unit Circle:

Center (0, 0)

radius = 1

x2 + y2 = 1

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Now using the definitions of the trig functions with r = 1, we have:

sin csc1

cos sec1

tan cot

1

1

yy

x

y y r

r y

xy x

x

x

y

x r

r x

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Find the value of the six trig functions for

Example*

2

(1, 0)

(0, 1)

(-1, 0)

(0, -1)

0

3

2

2

sin2

cos2

tan2

1csc

2

1sec

2

cot2

y

x

y

x

y

x

x

y

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Find the value of the six trig functions for

Example

7

sin 7

cos 7

tan 7

1csc 7

1sec 7

cot 7

y

x

y

x

y

xx

y

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Coterminal Angles

In each of these illustrations, angles and are coterminal.

is a negative angle

coterminal to is a positive angle (> 360°)

coterminal to

Two angles in standard position are said to be coterminal if they

have the same terminal sides.

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Example of Finding Coterminal Angles

You can find an angle that is coterminal to a given angle

by adding or subtracting multiples of 360º or 2.

Example:

Find one positive and one negative angle that are coterminal to 112º.

For a positive coterminal angle, add 360º : 112º + 360º = 472º

For a negative coterminal angle, subtract 360º: 112º - 360º = -248º

Note: There are an infinite number of angles that are coterminal to 112 º.

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Find one positive and one negative coterminal angle of 3

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Example

(a) sin 390 (b) cos 420

7sec

4

(c) csc 270 (d)

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I will use the notation to represent an angle’s reference angle.

The values of the trig functions for non-acute angles (Quads II, III, IV) can be found using the values of the corresponding reference angles.

Reference Angles

'

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Reference Angles

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Example

Find the reference angle for 225

Solution y

x

'

By sketching in standard position, we see that it is a 3rd quadrant angle. To find , you would subtract 180° from 225 °.

'

225 180'

' _____

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More Examples

Find the reference angles for the following angles.

1. 2. 3. 210 5

4

5.2

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So what’s so great about reference angles?

Well…to find the value of the trig function of any non-acute angle, we just need to find the trig function of the reference angle and then determine whether it is positive or negative, depending upon the quadrant in which the angle lies.

For example,

1sin 225 (sin 45 )

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45° is the ref angleIn Quad 3, sin is negative

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Trig Functions of Common Angles

Using reference angles and the special reference triangles, we can find the exact values of the common angles.

To find the value of a trig function for any common angle

1. Determine the quadrant in which the angle lies.

2. Determine the reference angle.

3. Use one of the special triangles to determine the function value for the reference angle.

4. Depending upon the quadrant in which lies, use the appropriate sign (+ or –).

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More Examples

Give the exact value of the trig function (without using a calculator).

1. 2.

5sin

6

3cos

4

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More Examples

Give the exact value of the trig function (without using a calculator).

3. 4.

cot 6604

csc3

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End of Section 7.4