Special Angles and their Trig Functions

By Jeannie Taylor

Through Funding Provided by a

VCCS LearningWare Grant

We will first look at the special angles called the quadrantal angles.

90

180

270

0

The quadrantal angles are those angles that lie on the axis of the Cartesian coordinate system: , , , and .0 90 180 270

We also need to be able to recognize these angles when they are given to us in radian measure. Look at the smallest possible positive angle in standard position, other than 0 , yet having the same terminal side as 0 . This is a 360 angle which is equivalent to .

radians2

radians2360

90

180

270

0

2

radians

If we look at half of that angle, we have

radiansor180

.

radians

Looking at the angle half-way between 180 and 360 , we have 270 or radians which is of the total (360 or ).

2

34

3

2 radians

Moving all the way around from 0 to 360 completes the circle and and gives the 360 angle which is equal to radians. 2

radians2

3

Looking at the angle half-way between 0 and 180 or , we have 90 or .

2

We can count the quadrantal angles in terms of .radians2

radians2

0 radiansradians2

2

radians2

3

radians2

4

Notice that after counting these angles based on portions of the full circle, two of these angles reduce to radians with which we are familiar, .

2 and

radians

radians2

Add the equivalent degree measure to each of these quadrantal angles. 0

90

180

270

radians57.1

radians14.3

radians71.4

radians28.6

We can approximate the radian measure of each angle to two decimal places. One of them, you already know, . It will probably be a good idea to memorize the others. Knowing all of these numbers allows you to quickly identify the location of any angle.

radians14.3

360

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

begin with the point (1, 0) on the x axis.

(1, 0)

radians2

0 radians

radians2

3

radians

radians2

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.

y

x

x

y

x

r

r

x

y

r

r

y

cottan

seccos

cscsin

For the angle 0 , we can see that x = 1 and y = 0. To visualize the length of r, think about the line of a 1 angle getting closer and closer to 0 at the point (1, 0).

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

radians2

0 radians

radians2

3

radians

radians2

0

90

180

270

360

(1, 0)

or

undefinedis0cot01

00tan

10sec10cos

undefinedis0csc00sin

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

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

Now let’s cut each quadrant in half, which basically gives us 8 equal sections.

0

4

4

2

4

4

4

6

4

3

4

5

4

7

4

8

The first angle, half way between 0 and would be .

2

422

1

We can again count around the circle, but this time we will count in terms of radians. Counting we say:

4

.4

8,

4

7,

4

6,

4

5,

4

4,

4

3,

4

2,

4

1 and

4

2

2

2

3

Then reduce appropriately.

45

90

135

180

225

270

315

360

Since 0 to radians is 90 , we know that is half of 90 or 45. Each successive angle is 45 more than the previous angle. Now we can name all of these special angles in degrees.

2

4

2

It is much easier to construct this picture of angles in both degrees and radians than it is to memorize a table involving these angles (45 or reference angles,).

4

45

45The lengths of the legs of the

45 – 45 – 90 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.

Next we will look at two special triangles: the 45 – 45 – 90 triangle and the 30 – 60 – 90 triangle. These triangles will allow us to easily find the trig functions of the special angles, 45 , 30 , and 60 .

45

45

1

1

2

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 trig functions for a angle.45

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

60

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 30, 60, and 90 .

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.

3

60

1

2

30

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 30 angle and a 60 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

45

45

1

12

Either memorizing or learning how to construct these triangles is much easier than memorizing tables for the 45 , 30 , and 60 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.

3

60

1

2

30

45

45

1

12

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.

y

x

Example 1: Find the six trig functions of 330 .

Second, find the reference angle, 360 - 330 = 30 First draw the 330 degree angle.

To compute the trig functions of the 30 angle, draw the “special” triangle.

3

60

1

2

30

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

AS

T C

330

30

y

x

330

3

60

1

2

30

AS

T C

3330cot3

3

3

1330tan

3

32

3

2330sec

2

3330cos

2330csc2

1330sin

Example 1 Continued: The six trig functions of 330 are:

30

y

x

Example 2: Find the six trig functions of . (Slide 1)

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 . (Slide 2)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

0 radians

Example 3: Find the exact value of cos .

4

5

We will first draw the angle to determine the quadrant.

4

5

4

4

2

4

3

4

4

We see that the angle is located in the 2nd quadrant and the cos is negative in the 2nd quadrant.

4

5

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

4

Note that the reference angle is .

4

4

5cos = 2

2

2

1

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 cos is negative in the 3rd quadrant.

4

13

cos =

4

132

2

2

1

4

12

4

13

4