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 ).
The graph of the trigonometry function special angles example1
The solution of the trigonometry function example1
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 ).
The graph of trigonometry function example2
The solution of the trigonometry function special angle example2
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 graph of the trigonometry function special angle example3
The solution of the trigonometry function special angle example3
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.
Trigonometry function special angles formula
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
The solution of the trigonometry function spexial angle example4
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 0o 90o 180o 270o 360o
a in radius 0 pi/2 pi 3pi/2 2pi
sin a 0 1 0 -1 0
cos a 1 0 -1 0 1
tan a 0 no exist 0 no exist 0