Download - Trig Functions of Special Angles
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Special Angles and their Trig Functions
By Jeannie Taylor
Through Funding Provided by a
VCCS LearningWare Grant
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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
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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
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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
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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, .
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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.
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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
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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
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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 .
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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
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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.
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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.
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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
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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