1. 7.5 The Other Trigonometric Functions
Objective To find values of the tangent, cotangent, secant, and cosecant functions and to sketch the functions’ graphs.

2. The Other Trigonometric Functionsy x
P (x, y)
(r, 0) r
Besides the sine and cosine functions, there are some other trigonometric functions. Define:
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.

3. Since we can write these other four
new functions in terms of sin and cos.
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.

4. The Domain of the Trigonometric Functions
The domain of the sine function is the set of all real numbers.
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 cotangent function is the set of all real numbers except integral multiples of  (180o).
The domain of the secant function is the set of all real numbers except odd multiples of /2 (90o).
The domain of the cosecant function is the set of all real numbers except integral multiples of  (180o).

5. 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).

6. Remember the six trigonometric functions defined using a point (x, y) on the terminal side of an angle,  .
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)
0 radians or
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).
As this line falls on top of the x-axis, we can see that the length of r is 1.

7. 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.
0 radians
(1, 0) or
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.

8. (0, 1)
0 radians
radians or
(-1, 0)
(0, -1)

9. 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 .
The lengths of the legs of the 45o – 45o – 90o triangle are equal to each other because their corresponding angles are equal. 45 1
If we let each leg have a length of 1, then we find the hypotenuse to be using the Pythagorean theorem. 1
You should memorize this triangle or at least be able to construct it. These angles will be used frequently.

10. 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. 45 1 1

11. 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. 2 2 1 1
We find the length of the other leg to be , using the Pythagorean theorem.
You should memorize this triangle or at least be able to construct it. These angles, also, will be used frequently.

12. 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. 2 2 1 1

13. 45 2 1 2 1 1 1
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.

14. The Special Values of All Trigonometric Functions

15. The 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.
Sine II
All I
A good way to remember this chart is that ASTC stands for All Students Take Calculus.
III
Tangent IV
Cosine

16. A S T C Example 1: Find the six trig functions of 330o .
[Solution] First draw the 330o angle.
Second, find the reference angle, 360o – 330o = 30o
To compute the trig functions of the 30o angle, draw the “special” triangle or recall from the table. x A S T C
Determine the correct sign for the trig functions of 330o . Only the cosine and the secant are “+”.
330o
30o

17. Example 1: Find the six trig functions of 330o .
[Solution] The six trig functions of 330o are:

18. Example 2: Find the six trig functions of .
First determine the location of
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 y x
We can see that the reference angle is , which is the same as Therefore, we will compute the trig functions of using the 60 angle of the special triangle. 1 2

19. A S T C Example 2: Find the six trig functions of .
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.. y x A S T C 1 2

20. Practice Exercises
Find the value of the sec 360 without using a calculator.
Find the exact value of the tan 420 .
Find the exact value of sin
Find the tan 270 without using a calculator.
Find the exact value of the csc
Find the exact value of the cot (-225 ).
Find the exact value of the sin
Find the exact value of the cos
Find the value of the cos(- ) without using a calculator.
Find the exact value of the sec 315 .

21. Key For The Practice Exercises
sec 360 = 1
tan 420 =
sin =
tan 270 is undefined
csc =
cot (-225 ) = -1
sin =
cos =
cos(- ) = -1
sec =
Problems 3 and 7 have solution explanations following this key.

22. A S T C Problem 3: Find the sin .
We will first draw the angle by counting in a negative direction in units of .
0 radians A S T C
We can see that is the reference angle and we know that is the same as So we will draw our 30 triangle and see that the sin 30 is .
All that’s left is to find the correct sign. 1 2
And we can see that the correct sign is “-”, since the sin is always “-” in the 3rd quadrant.
Answer: sin =

23. A S T C Problem 7: Find the exact value of cos .
We will first draw the angle to determine the quadrant.
We see that the angle is located in the 3rd quadrant and the cosine is negative in the 3rd quadrant.
0 radians A S T C
Note that the reference angle is .
We know that is the same as 45 , so the reference angle is Using the special triangle we can see that the cos of 45 or is . 45 1
cos =

24. 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.
If  is an 2nd quadrant angle, we can draw a diagram as shown at the right. Then:

25. Example 3: Given that tan = –3/4 , find the values of the other five trigonometric functions.
[Solution] If  is a 4th quadrant angle, we can draw a diagram as shown at the right. Then:
(4, -3) y x 5  -3 4

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

27. The Tangent Graph
Vertical Asymptote:  = k + /2, where k  Z

28. The Cotangent Graph
Vertical Asymptote:  = k, where k  Z

29. The Secant Graph

30. The Secant Graph Vertical Asymptote:  = k + /2, where k  Z
tan and sec have the same Vertical Asymptote:  = k + /2, where k  Z

31. The Cosecant Graph Vertical Asymptote:  = k, where k  Z
cot and csc have the same Vertical Asymptote:  = k , where k  Z

32. Periodic Properties
From the graphs of all these six trigonometric functions, we can easily see the following periodic properties:

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

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