7.5 The Other Trigonometric Functions Objective To find values of the tangent, cotangent, secant,...

7.5 The Other Trigonometric Functions7.5 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. Define:

tan , 0y

xx

cot , 0x

yy

sec , 0r

xx

csc , 0r

yy

From the above definitions, we notice that trig function tan and sec have the same domain, the terminal side of the angle can not be on y-axis. From the above definitions, we notice that trig function cot and csc have the same domain, the terminal side of the angle can not be on x-axis.

Since we can write these other four

new functions in terms of sin and cos.

cos and sin , x y

r r

/ sintan

/ cos

y y r

x x r

/ cos

cot/ sin

x x r

y y r

/ 1sec

/ cos

r r r

x x r

/ 1csc

/ sin

r r r

y y r

Notice that sec and cos are reciprocals, as are csc and sin .as are tan and cot .Also notice that after we rewrite the expression of other trig functions, the domain of these four new trig functions do not change. For example, from x 0 for tan , switches to cos 0, which is equivalent to the terminal side of the angel can not be on y-axis. As are the other trig functions.

The domain of the cosecant function is the set of all real numbers except integral multiples of (180o).

The domain of the cosine function is the set of all real numbers.

The domain of the tangent function is the set of all real numbers except odd multiples of /2 (90o).

The domain of the secant function is the set of all real numbers except odd multiples of /2 (90o).

The Domain of the Trigonometric Functions

The domain of the cotangent function is the set of all real numbers except integral multiples of (180o).

The domain of the sine function is the set of all real numbers.

/ 1sec

/ cos

r r r

x x r

/ 1csc

/ sin

r r r

y y r

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).

sin , cos y x

r r

We can find the trigonometric functions of the quadrantal angles using this definition. We will use

the unit circle with the point (1, 0) on the x-axis.

(1, 0)

radians2

0 radians

radians2

3

radian

2 radians

0

90

180

270

360or

As this line falls on top of the x-axis, we can see that the length of r is 1.

sin csc

cos sec

tan cot

y r

r y

x r

r xy x

x y

For the angle 0o , we use point (1, 0). To visualize the length of r, think about the line of a 1o angle getting closer and closer to 0o at the point (1, 0).

Remember the six trigonometric functions defined using a point (x, y) on the terminal side of an angle, .

sin 0 0, csc0 is

cos0 1, sec0 1

0tan 0 0, cot 0 is

1

undefined

undefined

Using the values, x = 1, y = 0, and r = 1, we list the six trig functions of 0o. And of course, these values also apply to 0o radians, 360o , 2 radians, etc.

It will be just as easy to find the trig functions of the remaining quadrantal angles using the point (x, y) and the r value of 1.

(1, 0)

radians2

0 radians

radians2

3

radian

2 radians

0

90

180

270

360or

radians2

0 radians

radians2

3

radians

radians2

0

90

180

270

360or

(0, 1)

02

cotundefinedis2

tan

undefinedis2

sec02

cos

12

csc12

sin

(-1, 0)undefinediscot0tan

1sec1cos

undefinediscsc0sin

(0, -1)

02

3cotundefinedis

2

3tan

undefinedis2

3sec0

2

3cos

12

3csc1

2

3sin

The lengths of the legs of the 45o – 45o – 90o triangle are equal to each other because their corresponding angles are equal.

If we let each leg have a length of 1, then we find the hypotenuse to be using the Pythagorean theorem.

2

1

1

2

You should memorize this triangle or at least be able to construct it. These angles will be used frequently.

Since we have learned that the reference angle plays an important role in evaluating the trigonometric functions, the next we will look at two special triangles: the 45o – 45o – 90o triangle and the 30o – 60o – 90o triangle. These triangles will allow us to easily find the trig functions of the special angles, 30o , 45o , and 60o .

45

45

145cot145tan

245sec2

245cos

245csc2

2

2

145sin

Using the definition of the trigonometric functions as the ratios of the sides of a right triangle, we can now list all six trigonometric functions for a 45o angle.

1

1

245

45

For the 30o – 60o – 90o triangle, we will construct an equilateral triangle (a triangle with 3 equal angles of 60o each, which guarantees 3 equal sides).

If we let each side be a length of 2, then cutting the triangle in half will give us a right triangle with a base of 1 and a hypotenuse of 2. This smaller triangle now has angles of 30o, 60o, and 90o.

We find the length of the other leg to be , using the Pythagorean theorem.

3

3

60

1

2

30

You should memorize this triangle or at least be able to construct it. These angles, also, will be used frequently.

30

60

1

2

Again, using the definition of the trigonometric functions as the ratios of the sides of a right triangle, we can now list all the trig functions for a 30o angle and a 60o angle.

330cot3

3

3

130tan

3

32

3

230sec

2

330cos

230csc2

130sin

3

3

3

160cot360tan

260sec2

160cos

3

32

3

260csc

2

360sin

3

60

1

2

30 30

60

1

2

Either memorizing or learning how to construct these triangles is much easier than memorizing tables for the 30o, 45o, and 60o angles. These angles are used frequently and often you need exact function values rather than rounded values. You cannot get exact values on your calculator.

1

1

245

45

3

60

1

2

30 30

60

1

2

The Special Values of All Trigonometric FunctionsThe Special Values of All Trigonometric Functions

The Sgin of All Trigonometric FunctionsThe Sgin of All Trigonometric Functions

Knowing these triangles, understanding the use of reference angles, and remembering how to get the proper sign of a function enables us to find exact values of these special angles. All

I

Sine

II

III

Tangent

IV

Cosine

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

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

60

1

2

30

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

3

60

1

2

30

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

Problems 3 and 7 have solution explanations following this key.

0 radians

Problem 3: Find the sin .

All that’s left is to find the correct sign.

And we can see that the correct sign is “-”, since the sin is always “-” in the 3rd quadrant.

AS

T C

6

5

6

6

2

6

36

4

6

5

We will first draw the angle by counting in a negative direction in units of .

6

We can see that is the reference angle and we know that is the same as 30 . So we will draw our 30 triangle and see that the sin 30 is .

6

6

2

1

3

60

1

2

30

Answer: sin =

6

52

1

6

0 radians

Problem 7: Find the exact value of cos .

We will first draw the angle to determine the quadrant.

AS

T C

45

45

1

12

We know that is the same as 45 , so the reference angle is 45 . Using the special triangle we can see that the cos of 45 or is .

2

1

4

4

Note that the reference angle is .

4

4

13

4

4

2

4

4

4

64

5 4

7

4

8

4

94

10

4

114

3We see that the angle is located in the 3rd quadrant and the cosine is negative in the 3rd quadrant.

4

13

cos =

4

132

2

2

1

4

12

4

13

4

Example 3: Given that tan = –3/4 , find the values of the other five trigonometric functions.

[Solution] Since tan = –3/4 < 0, so is an 2nd or 4th quadrant angle.

cosx

r 4

5 1

seccos

1 54 45

siny

r

3

5

1csc

sin

5

3

1cot

tan

4

3

If is an 2nd quadrant angle, we can draw a diagram as shown at the right. Then:

Example 3: Given that tan = –3/4 , find the values of the other five trigonometric functions.

cosx

r

4

5 1

seccos

1 54 45

siny

r 3

5

1csc

sin

5

3

1cot

tan

4

3

[Solution] If is a 4th quadrant angle, we can draw a diagram as shown at the right. Then:

(4, -3)

y

x

5

-34

The Tangent GraphThe Tangent Graph

The domain of the tangent function is the set of all real numbers except odd multiples of /2 (90o).

The Tangent GraphThe Tangent Graph

Vertical Asymptote: = k + /2, where k Z

The Cotangent GraphThe Cotangent Graph

Vertical Asymptote: = k, where k Z

The Secant GraphThe Secant Graph

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

Periodic PropertiesFrom the graphs of all these six trigonometric functions, we can easily see the following periodic properties:

Theorem Even-Odd PropertiesFrom the graphs of all these six trigonometric functions, we can easily see the following even-odd properties:

Assignment

P. 285 # 1 – 7, 9, 13, 15, 17, 23, 25, 27