7-5 The Other Trigonometric Functions

Objective: To find values of the tangent, cotangent, secant, and

cosecant functions and to sketch the functions’ graphs

The Other Trigonometric FunctionsThe Other Trigonometric Functions

(0, r)

(-r, 0)

(0, -r)

t

y

xP (x, y)

(r, 0)

r

Besides the sine and cosine functions, there are some other trigonometric functions.

tan , 0yx

x

cot , 0x

yy

sec , 0rx

x

csc , 0r

yy

Other Trigonometric Functions

sec , 0r hypotenuse

xx adjacent

cot , 0x adjacent

xy opposite

csc , 0r hypotenuse

xy opposite

tangent

cotangent

secant

cosecant

tan , 0y opposite

xx adjacent

we can write these other four functions in terms of sin and cos.

cos x

r

/ sintan

/ cos

y y r

x x r

/ coscot

/ sin

x x r

y y r

/ 1sec

/ cos

r r r

x x r

/ 1csc

/ sin

r r r

y y r

sin ,

y

r

Reciprocals• Secant and cosine are reciprocals.

• Cosecant and sine are reciprocals.

• Cotangent and tangent are reciprocals.

As for the “sec” and “csc” functions, as a way to help keep them straight I think, the "s" doesn't go with "s" and the "c" doesn't go with "c" so if we want secant, it won't be the one that starts with an "s" so it must be the reciprocal of cosine. (have to just remember that tangent & cotangent go together but this will help you with sine and cosine).

Fill in your trig table

The Special Values of All Trigonometric The Special Values of All Trigonometric FunctionsFunctions

Signs of functions in quadrants

I II III IV

sin

csc

+ + - -

cos

sec

+ - - +

tan

cot

+ - + -

The Sign of All Trigonometric FunctionsThe Sign of All Trigonometric Functions

All

I

Sine

II

III

Tangent

IV

Cosine

A good way to remember this chart is that ASTC stands for All Students Take Calculus.

Find the value of each expressionwith a calculator

a) Tan 185˚

b) Cot 155˚

c) Csc (-1)

d) Sec 11

a)0.0875

b) -2.145

c) -1.188

d) 226.0

Degree Mode

Radian Mode

x

Example 1: Find the six trig functions of 330o .

Second, find the reference angle, 360o – 330o = 30o

[Solution] First draw the 330o angle.

To compute the trig functions of the 30o angle, draw the “special” triangle or recall from the table.

Determine the correct sign for the trig functions of 330o . Only the cosine and the secant are “+”.

AS

T C

330o30o

1sin 330 sin 30 csc330 2

2

3 2 2 3cos330 cos30 sec330

2 33

3 3tan 330 tan 30 cot 330 3

3 3

[Solution] The six trig functions of 330o are:

Example 1: Find the six trig functions of 330o .

y

x

Example 2: Find the six trig functions of . 3

4

First determine the location of .3

4

3

3

2

3

3

3

3

3

4

3

With a denominator of 3, the distance from 0 to radians is cut into thirds. Count around the Cartesian coordinate system beginning at 0

until we get to .

3

4

We can see that the reference angle is , which is the same as 60 . Therefore, we will compute the trig functions of using the 60 angle of the special triangle.

3

3

AS

T C

Example 2: Find the six trig functions of . 3

4

y

x

3

3

2

3

4

3

3

3

3

1

3

4cot3

3

4tan

23

4sec

2

1

3

4cos

3

32

3

2

3

4csc

2

3

3

4sin

Before we write the functions, we need to determine the signs for each function. Remember “All Students Take Calculus”. Since the angle, , is located in the 3rd quadrant, only the tangent and cotangent are positive. All the other functions are negative..

3

4

Practice Exercises

1. Find the value of the sec 360 without using a calculator.

2. Find the exact value of the tan 420 .

3. Find the exact value of sin .

4. Find the tan 270 without using a calculator.

5. Find the exact value of the csc .

6. Find the exact value of the cot (-225 ).

7. Find the exact value of the sin .

8. Find the exact value of the cos .

9. Find the value of the cos(- ) without using a calculator.

10. Find the exact value of the sec 315 .

6

5

6

11

3

7

4

13

Key For The Practice Exercises

1. sec 360 = 1

2. tan 420 =

3. sin =

4. tan 270 is undefined

5. csc =

6. cot (-225 ) = -1

7. sin =

8. cos =

9. cos(- ) = -1

10. sec 315 =

6

11

3

7

4

13

3

6

52

1

3

32

3

2

2

2

2

1

2

3

2

If and -90˚< <90˚, find the values of the other five trigonometric functions.

15

17csc

r

y

17

15

csc

1sin

15

17csc

Since sin <0 and -90˚< <90˚, is a fourth-quadrant angle. All fourth-quadrant angles have an x > 0.

x² + y² = r²

x = √17² - 15² = 8

8

17

cos

1sec

17

8cos

r

x

15

8

tan

1cot

8

15

8

15tan

x

y

Assignment

P. 285 # 2,4,6, 13-18, 20, 23-28Quiz tomorrow sine, cosine, & tangentTest Wednesday

Tangent Graph

Unit circle at 90˚ would be (0,1) so tan would be 1/0. Is this possible?

Tangent Graph in Radians

The Cotangent GraphThe Cotangent Graph

Vertical Asymptote: = k, where k Z

The Secant GraphThe Secant Graph

Vertical Asymptote: = k + /2, where k Ztan and sec have the same Vertical Asymptote: = k + /2, where k Z

The Cosecant GraphThe Cosecant Graph

Vertical Asymptote: = k, where k Zcot and csc have the same Vertical Asymptote: = k , where k Z