How to find negative cosine - ACT Math

Based on this data, we can make a little triangle that looks like:

This is because .

Now, this means that must equal . (Recall that the cosine function is negative in the second quadrant.) Now, we are looking for:

or . This is the cosine of a reference angle of:

Looking at our little triangle above, we can see that the cosine of is .

Recall that when you calculate a trigonometric function for an obtuse angle like this, you always use the -axis as your reference point for your angle. (Hence, it is called the "reference angle.")

Now, it is easiest to think of this like you are drawing a little triangle in the third quadrant of the Cartesian plane. It would look like:

So, you first need to calculate the hypotenuse. You can do this by using the Pythagorean Theorem, , where and are the lengths of the legs of the triangle and the length of the hypotenuse. Rearranging the equation to solve for , you get:

Substituting in the given values:

So, the cosine of an angle is:

or, for your data, .

This is approximately . Rounding, this is . However, since is in the third quadrant your value must be negative: .

If the point to be reached is , then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that:

Thus,

Now, SOHCAHTOA tells us that , so we know that:

Thus, our cosine is approximately . However, as we are in the third quadrant, cosine must be negative! Therefore, our true cosine is .

If the point to be reached is , then we may envision a right triangle with sides and , and hypotenuse . The Pythagorean Theorem tells us that , so we plug in and find that: .

Thus, .

Now, SOHCAHTOA tells us that , so we know that:

Thus, our cosine is approximately . However, as we are in the second quadrant, cosine must be negative! Therefore, our true cosine is .