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The Value of Special Angles of Trigonometric Functions

Example 1
The terminal side of angle a is pi/6. The radius of the circle is one, that is, r = 1.The P ( x , y ) is the intersection of the circle and the terminal side . Find the value of x and y, sin a , cos a , and tan a.
Solution
In right triangle OPM, since angle a = pi/6 = 30o, so the length of its opposite is one-half of the hypotenuse, that is, PM = r/2 , since r = 1 (given), so PM = 1/2. Since sin a = PM/r = PM/1 = PM, so PM is called the line of sin a. Since cos a = OM/r = OM/1 = OM, so OM is called the line of cos a. Since tan a = AT/OA = AT/1 = AT, so AT is called the line of tan a. (Note: OA = OP = r = 1 ).
Example 2
The terminal side of angle a is pi/4. Find the value of x and y, sin a , cos a , and tan a.
Solution
In right triangle POM, since angle a = pi/4 = 45o, so its both legs have the same length. That is, PM = OM. Since r = 1, so PM = OM = (square root 2)/2. PM is called the line of sin a. OM is called the line of cos a. AT is called the line of tan a. (Note: OA = OP = r = 1 ).
Example 3
The terminal side of angle a is pi/3. Find the value of x and y, sin a , cos a , and tan a.
Solution
In right triangle POM, since angle a = pi/3 = 60o, so angle OPM = 30o, then the length of OM is one-half of the hypotenuse, that is, OM = r/2 , since r = 1 (given), so OM = 1/2. MP is called the line of sin a. OM is called the line of cos a. AT is called the line of tan a. (Note: OA = OP = r = 1 ).
The Value of Special Angles of Sine, Cosine, and Tangent
The following are the values of the special angles of sine, cosine, and tangent.
a in degree a in radius sin a cos a tan a
0o 0 0 1 0
30o pi/6 1/2 (square root of 3)/2 (square root of 3)/3
45o pi/4 (square root of 2)/2 (square root of 2)/2 1
60o pi/3 (square root of 3)/2 1/2 square root of 3
90o pi/2 1 0 no exist
180o pi 0 -1 0
Trigonometric Functions of Any Angles
The following are the formula of trigonometric functions of any angles.
k is a integer, k = ... -5,-4,-3,-2,-1,0,1,2,3,4,5 ...
Example 4
Find the value of sin 960o.
Solution
Formula used:
sin(a + k × 360o) = sin(a)
sin(180o + a) = - sin(a)
The terminal side of the angle a is in quadrant III, so sin(a) is < 0.
Example 5
Find the value of cos ( - 600o).
Solution
cos(-600o)
= cos(-2 × 360o + 120o)
= cos(120o)
= cos(180o - 60o)
= - cos(60o)
= -1/2
Formula used:
cos(a + k × 360o) = cos a
cos(180o - a) = - cos a
The terminal side of the angle a is in quadrant II, so cos(a) is < 0.
Example 6
Find the value of tan 945o + sin 510o.
Solution
tan945o + sin510o
= tan(2 × 360o + 225o) + sin(1 × 360o + 150o)
= tan225o + sin150o
= tan(180o + 45) + sin(180o - 30o)
= tan45o + sin30o
= 1 + 1/2
= 3/2
Formula used:
tan(a + k × 360o) = tan a
In quadrant III, tan(a) > 0
In quadrant II, sin(a) > 0
Sine, Cosine, and Tangent of Special Angles
The following are sine, cosine, and tangent of special angles whose terminal side coincides with x-axis or y-axis.
a in degree a in radius sin a cos a tan a 0o 90o 180o 270o 360o 0 pi/2 pi 3pi/2 2pi 0 1 0 -1 0 1 0 -1 0 1 0 no exist 0 no exist 0