### All SAT II Math II Resources

## Example Questions

### Example Question #1 : Finding Sides With Trigonometry

The area of a regular pentagon is 1,000. Give its perimeter to the nearest whole number.

**Possible Answers:**

**Correct answer:**

A regular pentagon can be divided into ten congruent triangles by its five radii and its five apothems. Each triangle has the following shape:

The area of one such triangle is , so the area of the entire pentagon is ten times this, or .

The area of the pentagon is 1,000, so

Also,

, or equivalently, , so we solve for in the equation:

The perimeter is ten times this, or 121.

### Example Question #1 : Finding Sides With Trigonometry

The area of a regular dodecagon (twelve-sided polygon) is 600. Give its perimeter to the nearest whole number.

**Possible Answers:**

**Correct answer:**

A regular dodecagon can be divided into twenty-four congruent triangles by its twelve radii and its twelve apothems, each of which is shaped as shown:

The area of one such triangle is , so the area of the entire dodecagon is twenty-four times this, or

.

The area of the dodecagon is 600, so

, or

.

Also,

, or equivalently, , so solve for in the equation

Solve for :

The perimeter is twenty-four times this:

### Example Question #1 : Finding Sides With Trigonometry

The area of a regular nonagon (nine-sided polygon) is 900. Give its perimeter to the nearest whole number.

**Possible Answers:**

**Correct answer:**

A regular nonagon can be divided into eighteen congruent triangles by its nine radii and its nine apothems, each of which is shaped as shown:

The area of one such triangle is , so the area of the entire nonagon is eighteen times this, or . Since the area is 900,

, or

.

Also,

, or equivalently, , so solve for in the equation

The perimeter of the nonagon is eighteen times this:

, the correct response.

### Example Question #1 : Finding Sides With Trigonometry

A *nonagon *is a nine-sided polygon.

Nonagon has diagonal with length 10. To the nearest tenth, give the length of one side.

**Possible Answers:**

**Correct answer:**

Construct the nonagon with diagonal .

We will concern ourselves with finding the length of .

Since , and is isosceles, then

The following diagram is formed (limiiting ourselves to ):

By the Law of Sines,

### Example Question #1 : Finding Sides With Trigonometry

Give the length of one side of a regular pentagon whose diagonals measure 10 each. (Nearest tenth)

**Possible Answers:**

**Correct answer:**

Construct the pentagon with diagonal .

We will concern ourselves with finding the length of .

Since , and is isosceles, then

The following diagram is formed:

By the Law of Sines,

### Example Question #1 : Finding Sides With Trigonometry

In :

Evaluate to the nearest whole unit.

**Possible Answers:**

**Correct answer:**

The Law of Sines states that given two angles of a triangle with measures , and their opposite sides of lengths , respectively,

,

or, equivalently,

.

, whose length is desired, and , whose length is given, are opposite and , respectively, so, in the sine formula, set , , , and in the Law of Sines formula, then solve for :

### Example Question #1 : Finding Sides With Trigonometry

Suppose the distance from a student's eyes to the floor is 4 feet. He stares up at the top of a tree that is 20 feet away, creating a 30 degree angle of elevation. How tall is the tree?

**Possible Answers:**

**Correct answer:**

The height of the tree requires using trigonometry to solve. The distance of the student to the tree , partial height of the tree , and the distance between the student's eyes to the top of the tree will form the right triangle.

The tangent operation will be best used for this scenario, since we have the known distance of the student to the tree, and the partial height of the tree.

Set up an equation to solve for the partial height of the tree.

Multiply by 20 on both sides.

We will need to add this with the height of the student's eyes to the ground to get the height of the tree.

The answer is:

### Example Question #1 : Finding Sides With Trigonometry

A *decagon *is a ten-sided polygon.

Decagon has diagonal with length 10. To the nearest tenth, give the length of one side.

**Possible Answers:**

**Correct answer:**

Construct the decagon with diagonal .

We will concern ourselves with finding the length of .

Since , and is isosceles, then

The following diagram is formed (limiting ourselves to ):

By the Law of Sines,

### Example Question #1 : Trigonometry

In :

Evaluate to the nearest degree.

**Possible Answers:**

**Correct answer:**

The figure referenced is below:

By the Law of Cosines, the relationship of the measure of an angle of a triangle and the three side lengths , , and , the sidelength opposite the aforementioned angle, is as follows:

All three side lengths are known, so we are solving for . Setting

, the length of the side opposite the unknown angle;

;

;

and ,

We get the equation

Solving for :

Taking the inverse cosine:

,

the correct response.

### Example Question #1 : Trigonometry

If , what is if is between and ?

**Possible Answers:**

**Correct answer:**

Recall that .

Therefore, we are looking for or .

Now, this has a reference angle of , but it is in the third quadrant. This means that the value will be negative. The value of is . However, given the quadrant of our angle, it will be .

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