# ACT Math : How to find the cosine of an angle

## Example Questions

### Example Question #1 : How To Find The Cosine Of An Angle

If , where  and ,

then what is ?

Explanation:

In the triangle below, the tangent of , or the opposite side of the angle divided by the adjacent side of the angle. According to the Pythagorean

Theorem, the

Thus the hypotenuse equals .

The cosine of an angle is the adjacent side of the angle divided by the hypotenuse of the triangle, giving us .

However, since  is , and when  is between is positive while  is negative. Thus,  is negative, giving us the final answer of .

### Example Question #3041 : Act Math

What is cos θ?

Explanation:

To get the radical out of the denominator, we multiply:

### Example Question #1 : How To Find The Cosine Of An Angle

What is the cosine of the angle formed between the -axis and the line passing through  with a slope of ?  Round to the nearest hundredth.

Explanation:

You do not even need to compute the line for this question.  All that you need to do is note that you can make a little right triangle with a height of  and a base of , which you get from the slope of the line.  So, to compute the cosine, you will need to find the hypotenuse using the Pythagorean theorem:

So, our little triangle looks like:

The cosine will be the adjacent side, , divided by the hypotenuse, :

, or approximately .

### Example Question #122 : Trigonometry

What is the  of an angle with ?

Explanation:

Since the sine of the angle is , that means the opposite side of the triangle is  and the hypotenuse is . Automatically, you know this is a special 3-4-5 right triangle, and that the missing side is 3. If not, you could also find the 3rd side by doing Pythagorean Theorem. This gives you your answer of

### Example Question #3045 : Act Math

Right triangle  has sides  and . What is the cosine of

Explanation:

SOHCAHTOA tells us that , and we know the hypotenuse is the longest side of the triangle, . Our adjacent side will be the other side that has  as a vertex, .

Thus, .

### Example Question #3046 : Act Math

Right triangle  has sides  and . What is the cosine of