### All HiSET: Math Resources

## Example Questions

### Example Question #1 : Tangent

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

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

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 #2 : Tangent

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

**Possible Answers:**

None of the other choices gives the correct response.

**Correct answer:**

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 #3 : Tangent

Evaluate in terms of .

**Possible Answers:**

**Correct answer:**

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

.

### Example Question #4 : Tangent

**Possible Answers:**

**Correct answer:**

The sine of an angle is defined to be the ratio of the length of the opposite leg of a right triangle to the length of its hypotenuse. Therefore, we can set . By the Pythagorean Theorem:

the adjacent leg of the triangle has measure

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

,

the correct response.

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