SAT II Math II : Finding Sides with Trigonometry

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.

Explanation:

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 : Trigonometry

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

Explanation:

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.

Explanation:

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

nonagon is a nine-sided polygon.

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

Explanation:

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)

Explanation:

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 #6 : Finding Sides With Trigonometry

In :

Evaluate  to the nearest whole unit.

Explanation:

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?

Explanation:

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.

Example Question #1 : Finding Sides With Trigonometry

decagon is a ten-sided polygon.

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

Explanation:

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,