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SAT II Math II : Finding Sides with Trigonometry

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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:

107

199

121

118

Correct answer:

121

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:

Thingy_1

The area of one such triangle is $\frac{1}{2}ab$ , so the area of the entire pentagon is ten times this, or $10 \cdot \frac{1}{2}ab = 5ab$ .

The area of the pentagon is 1,000, so

5ab = 1,000

ab = 200

Also,

$\frac{a}{b} = \tan 54^{\circ }$ , or equivalently, $a=b \tan 54^{\circ }$ , so we solve for in the equation:

$b \tan 54^{ \circ } \cdot b = 200$

$b^{2} \tan 54^{ \circ } = 200$

$b^{2} = \frac{200}{\tan 54^{ \circ }} \approx \frac{200}{ 1.3764} \approx 145.3$

$b \approx \sqrt{145.3} \approx 12.1$

The perimeter is ten times this, or 121.

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Example Question #1 : Trigonometry

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

Possible Answers:

129

102

147

Correct answer:

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:

Thingy_2

The area of one such triangle is $\frac{1}{2}ab$ , so the area of the entire dodecagon is twenty-four times this, or

$24 \cdot \frac{1}{2}ab = 12ab$ .

The area of the dodecagon is 600, so

12ab = 600 , or

ab = 50 .

Also,

$\frac{a}{b} = \tan 75^{\circ }$ , or equivalently, $a=b \tan 75^{\circ }$ , so solve for in the equation

$b \tan 75^{ \circ } \cdot b =50$

Solve for :

$b^{2} \cdot \tan 75^{ \circ } = 50$

$b^{2} = \frac{50}{ \tan 75^{ \circ }} \approx \frac{50}{ 3.7321} \approx 13.4$

$b \approx \sqrt{13.4} \approx 3.66$

The perimeter is twenty-four times this:

$P \approx 3.66 \cdot 24 \approx 88$

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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:

137

109

165

Correct answer:

109

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:

Thingy_1

The area of one such triangle is $\frac{1}{2}ab$ , so the area of the entire nonagon is eighteen times this, or $18 \cdot \frac{1}{2}ab = 9ab$ . Since the area is 900,

9ab = 900 , or

ab= 100 .

Also,

$\frac{a}{b} = \tan 70^{\circ }$ , or equivalently, $a=b \tan 70^{\circ }$ , so solve for in the equation

$b \tan 70^{\circ } \cdot b= 100$

$b ^{2} \cdot \tan 70^{\circ } = 100$

$b ^{2} = \frac{100}{\tan 70^{\circ } } \approx \frac{100}{ 2.7475 } \approx 36.40$

$b \approx \sqrt{36.40} \approx 6.03$

The perimeter of the nonagon is eighteen times this:

$18 \cdot 6.03 \approx 109$ , the correct response.

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Example Question #4 : Finding Sides With Trigonometry

A nonagon is a nine-sided polygon.

Nonagon ABCDEFGHI has diagonal $\overline{AC}$ with length 10. To the nearest tenth, give the length of one side.

Possible Answers:

6.2

8.3

5.9

8.7

5.3

Correct answer:

5.3

Explanation:

Construct the nonagon with diagonal $\overline{AC}$ .

We will concern ourselves with finding the length of $\overline{BC}$ .

Since $m \angle ABC = \frac{(9-2)\cdot 180^{ \circ}}{9}= 140 ^{ \circ}$ , and $\Delta ABC$ is isosceles, then

$m \angle BAC = \frac{1}{2} (180^{\circ } - 140^{\circ }) = 20^{\circ }$

The following diagram is formed (limiiting ourselves to $\Delta ABC$ ):

Decagon

By the Law of Sines,

$\frac{BC}{\sin \left (m \angle BAC \right )} = \frac{AC} {\sin \left (m \angle ABC \right )}$

$\frac{BC}{\sin 20^{\circ }} = \frac{10} {\sin 140^{\circ }}$

$BC= \frac{10\sin 20^{\circ }} {\sin 140^{\circ }}$

$BC\approx \frac{10\cdot 0.3420} {0.6428}$

$BC \approx 5.3$

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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:

7.3

6.2

8.5

5.4

5.8

Correct answer:

6.2

Explanation:

Construct the pentagon with diagonal $\overline{AC}$ .

We will concern ourselves with finding the length of $\overline{BC}$ .

Since $m \angle ABC = \frac{(5-2)\cdot 180^{ \circ}}{5}= 108 ^{ \circ}$ , and $\Delta ABC$ is isosceles, then

$m \angle BAC = \frac{1}{2} (180^{\circ } - 108^{\circ }) = 36^{\circ }$

The following diagram is formed:

Pentagon

By the Law of Sines,

$\frac{BC}{\sin \left (m \angle BAC \right )} = \frac{AC} {\sin \left (m \angle ABC \right )}$

$\frac{BC}{\sin 36^{\circ }} = \frac{10} {\sin 108^{\circ }}$

$BC= \frac{10\sin 36^{\circ }} {\sin 108^{\circ }}$

$BC\approx \frac{10\cdot 0.5878} {0.9511}$

$BC \approx 6.2$

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Example Question #6 : Finding Sides With Trigonometry

In $\bigtriangleup ABC$ :

$m \angle A = 116 ^{\circ }$

$m \angle B = 20^{\circ }$

BC = 40

Evaluate to the nearest whole unit.

Possible Answers:

105

Correct answer:

Explanation:

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

$\frac{\sin \alpha}{a} = \frac{\sin \beta}{b}$ ,

or, equivalently,

$\frac{b}{\sin \beta} = \frac{a}{\sin \alpha}$ .

$\overline{AC }$ , whose length is desired, and $\overline{BC}$ , whose length is given, are opposite $\angle B$ and $\angle A$ , respectively, so, in the sine formula, set b = AC , $\beta = m \angle B = 20 ^{\circ }$ , a = BC = 40 , and $\alpha = m \angle A = 116^{\circ }$ in the Law of Sines formula, then solve for :

$\frac{AC}{\sin 20^{\circ }} = \frac{40}{\sin 116^{\circ }}$

$\frac{AC}{\sin 20^{\circ }} \cdot \sin 20^{\circ } = \frac{40}{\sin 116^{\circ }} \cdot \sin 20^{\circ }$

$AC = \frac{40}{\sin 116^{\circ }} \cdot \sin 20^{\circ }$

$\approx \frac{40}{0.8988} \cdot 0.3420$

$\approx 15$

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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:

$10\sqrt3+4$

$\frac{20\sqrt{3}+12}{3}$

$\frac{20\sqrt{3}+4}{3}$

Correct answer:

$\frac{20\sqrt{3}+12}{3}$

Explanation:

The height of the tree requires using trigonometry to solve. The distance of the student to the tree (20) , partial height of the tree (P) , 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.

$tan(\theta) = \frac{\textup{Opposite side to angle}}{\textup{Adjacent side to angle}}$

$tan(30) = \frac{P}{20}$

Multiply by 20 on both sides.

$P=20tan(30) = 20(\frac{\sqrt{3}}{3}) = \frac{20\sqrt{3}}{3}$

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

$\frac{20\sqrt{3}}{3} + 4 = \frac{20\sqrt{3}}{3} + \frac{12}{3}$

The answer is: $\frac{20\sqrt{3}+12}{3}$

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Example Question #1 : Finding Sides With Trigonometry

A decagon is a ten-sided polygon.

Decagon ABCDEFGHIJ has diagonal $\overline{AC}$ with length 10. To the nearest tenth, give the length of one side.

Possible Answers:

5.3

6.2

5.9

8.3

8.7

Correct answer:

5.3

Explanation:

Construct the decagon with diagonal $\overline{AC}$ .

We will concern ourselves with finding the length of $\overline{BC}$ .

Since $m \angle ABC = \frac{(10-2)\cdot 180^{ \circ}}{10}= 144 ^{ \circ}$ , and $\Delta ABC$ is isosceles, then

$m \angle BAC = \frac{1}{2} (180^{\circ } - 144^{\circ }) = 18^{\circ }$

The following diagram is formed (limiting ourselves to $\Delta ABC$ ):

Decagon

By the Law of Sines,

$\frac{BC}{\sin \left (m \angle BAC \right )} = \frac{AC} {\sin \left (m \angle ABC \right )}$

$\frac{BC}{\sin 18^{\circ }} = \frac{10} {\sin 144^{\circ }}$

$BC= \frac{10\sin 18^{\circ }} {\sin 144^{\circ }}$

$BC\approx \frac{10\cdot 0.3090} {0.5878}$

$BC \approx 5.3$

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SAT II Math II : Finding Sides with Trigonometry

Study concepts, example questions & explanations for SAT II Math II

All SAT II Math II Resources

Example Questions

Example Question #1 : Finding Sides With Trigonometry

Example Question #1 : Trigonometry

Example Question #1 : Finding Sides With Trigonometry

Example Question #4 : Finding Sides With Trigonometry

Example Question #1 : Finding Sides With Trigonometry

Example Question #6 : Finding Sides With Trigonometry

Example Question #1 : Finding Sides With Trigonometry

Example Question #1 : Finding Sides With Trigonometry

All SAT II Math II Resources

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