2005 iTest Problems/Problem 5

Problem

$\sin 30^\circ + \sin 45^\circ + \sin 60^\circ + \sin 90^\circ + \cos 120^\circ + \cos 135^\circ + \cos 150^\circ + \cos 180^\circ = ?$

Solution 1

The trigonometric values above can be quickly found on the unit circle, and are values that you should memorize. We will find one as an example, and the rest can be found by the reader. The sine of 30 degrees is the $y$ -coordinate of the point on the unit circle that makes an angle of $30^\circ$ with the horizontal. Dropping the altitude from that point to the x-axis gives us a $30-60-90$ right triangle. Since the hypotenuse is 1, the y-coordinate of the point is $0.5$ . Therefore, the sine of 30 degrees is $0.5$ . Repeating this process for the other values, we finally plug in. Our answer is $0.5+\frac{\sqrt{2}}{2}+\frac{\sqrt{3}}{2}+1-0.5-\frac{\sqrt{2}}{2}-1=\boxed{0}$ .

2005 iTest (Problems, Answer Key)
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2005 iTest Problems/Problem 5

Problem

Solution 1

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