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Special Angles

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The special angles are 30^\circ, 45^\circ, and 60^\circ. 90^\circ could also be considered a special angle. The values of the trigonometric functions of these angles have specific values. It would be best to commit these values to memory. Consequently, this lesson is mostly memorization.

First, it is useful to know the radian measures of these angles. In radian measure, the angles are equal to \pi divided by the number of times it goes into 180 since 180^\circ is equal to \pi in radian. So 30^\circ is the same as \frac{\pi}{6}, 45^\circ is the same as \frac{\pi}{4}, 60^\circ is the same as \frac{\pi}{3}, and 90^\circ is the same as \frac{\pi}{2}. Multiples of these angles’ radian measures are found by multiplying the radian measure by then number of times the measure goes into it. For example, 120^\circ, which is equal to 60^\circ \cdot 2, has a radian measure of \frac{2\pi}{3}.

The rest is memorization. To my knowledge, there is no other option; however, only the sine and cosine functions need to be remembered. The other functions can be found based off of these as shown in the last lesson. Without further ado, the sine of 30^\circ is \frac{1}{2} and the cosine is \frac{\sqrt{3}}{2}. Because sine and cosine are reciprocal functions, the values for 60^\circ are reversed. So the sine of 60^\circ is \frac{\sqrt{3}}{2} and the cosine is \frac{1}{2}. Also, the sine and cosine functions for 45^\circ are the same. Both equal \frac{1}{\sqrt{2}} which is \frac{\sqrt{2}}{2} rationalized.

Unit Circle Trig Functions

Remember that the cosine of the angle is the x-coordinate and the sine is the y-coordinate on the unit circle.

Added on April 6, 2008:

Angles whose reference angles is a special angle will also have these trigonometric values; however, they may be positive or negative depending on what quadrant it lies in. Since sine is related to the y-coordinate, it will be positive when the angle lies above the x-axis and negative when below. Since cosine is related to the x-coordinate, it will be positive when the angle lies to the right of the y-axis and negative when it lies to the left. As always, the other functions depend on these two.

Written by todizzle91

March 31, 2008 at 4:59 pm

Posted in Trigonometry

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  1. You can work out the value for 30 and 60 by considering an equilateral triangle with unit sides and the value for 45 with a right angled isosceles triangle using Pythagoras reasonably easily.

    David Woodford

    April 24, 2009 at 2:41 pm

    Reply

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