ACT Math : Cosine

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #3 : How To Find Negative Cosine

What is the cosine of the angle formed between the origin and the point  if that angle is formed with one side of the angle beginning on the -axis and then rotating counter-clockwise to ? Round to the nearest hundredth.

Possible Answers:

Correct answer:

Explanation:

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 second quadrant of the Cartesian plane. It would look like:

Cos37

So, you first need to  calculate the hypotenuse:

So, the cosine of an angle is:

  or, for your data, .  

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

Example Question #2 : How To Find Negative Cosine

To the nearest , what is the cosine of the angle formed between the origin and ? Assume a counterclockwise rotation.

Possible Answers:

Correct answer:

Explanation:

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 .

Example Question #3 : How To Find Negative Cosine

On a grid, what is the cosine of the angle formed between a line from the origin to  and the x-axis?

Possible Answers:

Correct answer:

Explanation:

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 .

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