# HiSET: Math : Tangent

## Example Questions

### Example Question #5 : Problems Involving Right Triangle Trigonometry

Refer to the triangle in the above diagram. Which of the following expressions correctly gives its area?

None of the other choices gives the correct response.

Explanation:

The area of a right triangle is half the product of the lengths of its legs, which here are  and  - that is,

We are given that  is the leg opposite the angle  and  is its adjacent leg, we can find  using the tangent ratio:

Setting  and , we get

Solve for  by multiplying both sides by 12:

Now, set  and  in the area formula:

,

the correct choice.

### Example Question #1 : Tangent

Refer to the triangle in the above diagram. Which of the following expressions correctly gives its area?

None of the other choices gives the correct response.

Explanation:

The area of a right triangle is half the product of the lengths of its legs, which here are  and  - that is,

We are given that  is the leg opposite the angle  and  is its adjacent leg, we can find  using the tangent ratio:

Setting  and , we get

Solve for  by first, finding the reciprocal of both sides:

Now, multiply both sides by 8:

Now, set  and  in the area formula:

,

the correct choice.

### Example Question #5 : Problems Involving Right Triangle Trigonometry

Evaluate  in terms of .

Explanation:

Suppose we allow be the lengths of the opposite leg and adjacent leg and the hypotenuse, respectively, of the right triangle with an acute angle measuring . The cosine is defined to be the ratio of the length of the adjacent side to that of the hypotenuse, so

We can set the lengths of the adjacent leg and the hypotenuse to and 3, respectively. By the Pythagorean Theorem, the length of the opposite leg is

The tangent of the angle is the ratio of the length of the opposite leg to that of the adjacent leg, so

.