ACT Math : How to find negative cosine

Study concepts, example questions & explanations for ACT Math

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Example Questions

Example Question #1 : How To Find Negative Cosine

If  and , what is the value of ?

Possible Answers:

Correct answer:

Explanation:

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

Rt1

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 .

Example Question #157 : Trigonometry

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

Cos611

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: .

Example Question #158 : Trigonometry

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 #159 : Trigonometry

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 #160 : Trigonometry

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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