Q: A 12 meter flagpole casts a 9 meter shadow. What is the angle of elevation of the sun from the ground? (Round to the nearest degree.)

$$\theta = \arctan \left( \dfrac{12}{9} \right ) \approx 53^\circ$$

Q: Evaluate $$\tan 35^\circ - 2 \cot 55^\circ$$ without a calculator.

$$\tan 35^\circ - 2 \cot 55^\circ = \tan 35^\circ - 2 \cot 55^\circ = -1$$

Question 1

Given a right triangle with legs $$a = 5$$ and $$b = 12$$, what is the exact value of $$\sin P$$ where $$P is $$the angle opposite side $$a$$?

Accepted Answer

$$\sin P = \dfrac{5}{13}$$

Question 2

In a right triangle, if $$a = 13$$ and $$c = 6$$, what is $$\cos \alpha$$?

Accepted Answer

$$\cos \alpha = \dfrac{12}{13}$$

Question 3

A 22-foot extension ladder leans against a building, making a $$70^\circ$$ angle with the ground. How high up the building does the ladder touch? (Round to one decimal place.)

Accepted Answer

$$22 \cdot \sin 70^\circ \approx 20.7$$ feet

Question 4

A 12 meter flagpole casts a 9 meter shadow. What is the angle of elevation of the sun from the ground? (Round to the nearest degree.)

Accepted Answer

$$	heta = \arctan \left( \dfrac{12}{9} ight ) \approx 53^\circ$$

Question 5

You fly a kite with 100 feet of string, and the string makes a $$75^\circ$$ angle with the ground. How tall is the tree where the kite is stuck? (Assume the string is straight and taut. Round to the nearest foot.)

Accepted Answer

$$100 \cdot \sin 75^\circ \approx 97$$ feet

Question 6

Find the exact value of $$\sin 20^\circ \cos 70^\circ.$$

Accepted Answer

$$\dfrac{1}{2} \sin 40^\circ$$

Question 7

Evaluate $$	an 35^\circ - 2 \cot 55^\circ$$ without a calculator.

Accepted Answer

$$	an 35^\circ - 2 \cot 55^\circ = 	an 35^\circ - 2 \cot 55^\circ = -1$$

Question 8

Given triangle $$ABC$$ with $$\alpha = 85^\circ$$, $$\gamma = 50^\circ$$, and $$b = 3$$, use the Law of Sines to find $$a ($$rounded to two decimal places).

Accepted Answer

$$a = \dfrac{3 \sin 85^\circ}{\sin 45^\circ} \approx 4.24$$

Question 9

Given triangle $$ABC$$ with $$a = 3$$, $$b = 7$$, and $$\alpha = 70^\circ$$, use the Law of Sines to find $$\beta ($$rounded to the nearest degree).

Accepted Answer

$$\beta = \arcsin \left( \dfrac{3 \sin 70^\circ}{7} \right ) \approx 25^\circ$$

Question 10

John stands 100 feet from the base of a tree that grows at an $$83^\circ$$ angle with the ground. If the angle of elevation from John to the top of the tree is $$33^\circ$$, how tall is the tree? (Round to the nearest foot.)

Accepted Answer

$$\dfrac{100 \sin 33^\circ}{\sin 64^\circ} \approx 56$$ feet

Question 11

Given triangle $$ABC$$ with $$a = 2$$, $$c = 5$$, and $$\beta = 20^\circ$$, use the Law of Cosines to find $$b ($$rounded to two decimal places).

Accepted Answer

$$b^2$$ = $$2^2$$ + $$5^2 - 2$$ \cdot 2 \cdot 5 \cos 20^\circ$$,\ b \approx 3.46$

Question 12

Given triangle $$ABC$$ with $$a = 4$$, $$b = 3$$, $$c = 6$$, use the Law of Cosines to find $$\alpha ($$rounded to the nearest degree).

Accepted Answer

$$\alpha = \arccos \left( \dfrac{4^2$ + 6^2$ - 3^2$}{2 \cdot 4 \cdot 6} \right ) \approx 70^\circ$$

Question 13

A baseball diamond is a square with sides of 90 feet. The distance from home plate to dead center field is 400 feet. How far is it from first base to dead center field? (Round to the nearest foot.)

Accepted Answer

$$\sqrt{(400^2$ + 90^2$ - 2 \cdot 400 \cdot 90 \cos 45^\circ)} \approx 318$$ feet

Question 14

Given triangle $$ABC$$ with $$a = 15$$, $$b = 17$$, $$c = 13$$, use the Law of Cosines to find $$\gamma ($$rounded to the nearest degree).

Accepted Answer

$$\gamma = \arccos \left( \dfrac{15^2$ + 17^2$ - 13^2$}{2 \cdot 15 \cdot 17} \right ) \approx 50^\circ$$

Question 15

Which of the following is the correct Law of Sines formula for solving for side $$a in $$triangle $$ABC$$?

Accepted Answer

$$\dfrac{a}{\sin \alpha} = \dfrac{b}{\sin \beta} = \dfrac{c}{\sin \gamma}$$

Question 16

Which scenario requires the Law of Cosines rather than the Law of Sines to solve a triangle?

Accepted Answer

Given all three sides ($$SSS)$$

Question 17

If $$	an 75^\circ - \cot 15^\circ = x$$, what is the value of $$x$$?

Accepted Answer

$$x = 0$$

Question 18

Given triangle $$ABC$$ with $$a = 3.44$$, $$\beta = 2^\circ$$, and $$c = 2.3$$, which law would you use to solve for $$b$$ and why?

Accepted Answer

Law of Cosines, because two sides and the included angle are given ($$SAS)$$

Precalculus Right Triangle and Law of Sines/Cosines Study Guide

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