Find the exact value of the sin of negative 120 degrees without using a calculator.
In order to answer this question, we will begin by sketching the unit circle in order to identify which quadrant the angle negative 120 degrees lies in. We know that any angle in standard position is measured from the positive 𝑥-axis. If the angle is positive, we measure in the counterclockwise direction, whereas if the angle is negative, as in this case, we measure in the clockwise direction. Marking on the angles negative 90, negative 180, negative 270, and negative 360 degrees, we see that our angle lies in the third quadrant, as negative 120 lies between negative 90 and negative 180.
We know that any point that lies on the unit circle has coordinates cos 𝜃, sin 𝜃. This means that the point at which the terminal side of our angle intersects the unit circle has coordinates cos negative 120 degrees, sin negative 120 degrees. It is the value of this 𝑦-coordinate, the sin of negative 120 degrees, that we’re trying to calculate. As our point lies in the third quadrant, we know that both the 𝑥- and 𝑦-coordinates will be negative. This means that the sin of negative 120 degrees is negative. In order to calculate its exact value, we will draw a perpendicular line from the 𝑥-axis to the point. This creates a right triangle as shown, where the length of the hypotenuse is one unit.
Next, we can use our knowledge of right angle trigonometry once we have calculated the value of the reference angle 𝛼. We know that the measure of the acute angle the terminal side makes with the 𝑥-axis is called the reference angle of 𝜃. Since the directed angle is negative, we know that the sum of the magnitude of this angle and 𝛼 must equal 180 degrees. This means that 𝛼 plus 120 degrees equals 180 degrees. Subtracting 120 degrees from both sides, we have 𝛼 is equal to 60 degrees. The reference angle, which is inside our right triangle, is equal to 60 degrees.
The side adjacent to this angle and the right angle has length equal to the absolute value of the cos of negative 120 degrees. And the side opposite our angle has length equal to the absolute value of the sin of negative 120 degrees. After clearing some space, we recall that the sine ratio tells us that the sin of any angle 𝜃 is equal to the opposite over the hypotenuse. This means that the sin of 60 degrees is equal to the absolute value of the sin of negative 120 degrees over one.
60 degrees is one of our special angles. And we know that the sin of 60 degrees is root three over two. This is therefore equal to the absolute value of the sin of negative 120 degrees. As we have already established that the sin of negative 120 degrees is negative, this must be equal to negative root three over two.