True or False — Without Using a Calculator: cos of negative 120 degrees is equal to cos of 60 degrees.
We begin by recalling that 60 degrees is one of our special angles. And the cos of 60 degrees is equal to one-half. This means that in order to answer this question, we need to work out whether the cos of negative 120 degrees is also equal to one-half. We will do this by sketching the unit circle, recalling that any point that lies on this circle with radius one has coordinates cos 𝜃, sin 𝜃.
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. By firstly marking negative 90 and negative 180 degrees, we can draw the angle negative 120 degrees in standard position as shown. The point at which the terminal side of our angle intersects the unit circle has coordinates cos negative 120 degrees, sin of negative 120 degrees. As this point lies in the third quadrant, it is clear that both the 𝑥- and 𝑦-coordinates will be negative. This means that the cos of negative 120 degrees must be negative. Since the cos of 60 degrees is positive, it is clear that the cos of negative 120 degrees is not equal to the cos of 60 degrees. And we can therefore conclude that the statement in the question is false.
Whilst it is not required in this question, we could go one stage further here and calculate the exact value of the cos of negative 120 degrees. This can be done using the triangle drawn in the third quadrant, together with our knowledge of right-angled trigonometry. The length of the side opposite the 60-degree angle is the absolute value of the sin of negative 120 degrees. The side adjacent to the 60-degree angle and the right angle has length equal to the absolute value of the cos of negative 120 degrees. The length of the hypotenuse, which is the radius of the circle, is equal to one unit.
The cosine ratio tells us that the cos of any angle 𝜃 in a right triangle is equal to the adjacent over the hypotenuse. This means that, in our triangle, the cos of 60 degrees is equal to the absolute value of the cos of negative 120 degrees over one. Since the cos of 60 degrees is one-half, the absolute value of the cos of negative 120 degrees is also equal to one-half. And since the cos of negative 120 degrees must be negative, this is equal to negative one-half. Whilst the two expressions in the question are not equal, they are the additive inverse of one another. However, in relation to this specific question, this confirms that the correct answer is false.