Multiples of 30 degrees

and

Reference angle = 240o - 180o = 60o

Example 2

Find the

First, we have to find the coterminal angle of

Divide the angle by 2p to compute the quotient k

Thus, k = 18

coterminal angle =

Figure 3 illustrate the computation of the sine and cosine of

Figure 3

Reference angle = p - 5p/6 = p/6

Math 002

A. Farhat

Finding Trigonometric Values of Multiples of 60 degrees (p/6 radians)

Figure 1 illustrates the construction of the 30-60 triangle to compute the values of the trigonometric functions for the special 30o and 60o angles. Let ABC be an equilateral triangle with sides equal to 1. The bisector AD of angle A also bisect the opposite side BC at the point D. Thus, the opposite side of the 30o angle BAD equals 1/2. The length of the side AD can be computed using Pythagorean theorem:

Figure 1: Construction of the 30-60 triangle

The special 30-60 triangle with hypotenuse 1 can be remembered as a triangle having a "short" side equals

, and a "long" side equals

. This triangle can be inscribed inside the unit circle and used to find the values of the cosine and the sine functions (and consequently the rest of the trigonometric functions) for any multiple of 30-degree angles. Figure 2 illustrates the division of the unit circle in terms of multiples of 30 degrees (p /6 radians). It helps to remember the quadrantal angles as 0, 90, 180, 270, and 360 degrees (0,

, and

radians).

Figure 2: Multiples of 30o (

)

Procedure for finding the values of sine and cosine

1. Draw a unit circle and locate the point on the unit circle representing the angle in question. (multiple of 30 degrees or

radians)

2. Draw a perpendicular line from the point to the x-axis and a line connecting the point to the center of the circle. This will form the 30-60 triangle.

3. Mark the "short" side of the triangle as 1/2 or -1/2 and mark the "long" side of the triangle as

(Depending on the location of the triangle)

4. The length of the side on the x-axis will be the cosine of the angle and the length of the side parallel to the y-axis will be the sine of the angle.

5 . The central angle formed by the triangle, which will be either 30o or 60o, represents the reference angle of the angle in question

Example 1

Find the sine and cosine of 240o

(In terms of multiples of

, the angle

)

The following graphs illustrate the steps to find the cosine and sine of the angle.

Locate the point corresponding

to the angle in question

Connect the point to the

x-axis and the center

Mark the length of the sides of the triangle (including the appropriate sign)

Thus,