Q: Evaluate $$\tan 35^\circ - 2 \cot 55^\circ$$ exactly.

$$\tan 35^\circ - 2 \cot 55^\circ = 0$$

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

$$\cos \alpha = \dfrac{13}{\sqrt{13^2$ + 6^2$}}$$

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

$$\arctan \left( \dfrac{9}{12} \right) \approx 37^\circ$$

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 taut and straight.)

Accepted Answer

$$100 \cdot \sin 75^\circ \approx 96.6$$ feet

Question 6

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

Accepted Answer

$$\dfrac{1}{2} \sin 90^\circ = \dfrac{1}{2}$$

Question 7

Evaluate $$	an 35^\circ - 2 \cot 55^\circ$$ exactly.

Accepted Answer

$$	an 35^\circ - 2 \cot 55^\circ = 0$$

Question 8

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

Accepted Answer

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

Question 9

Given triangle $$ABC$$ with $$a = 4$$, $$b = 2$$, find $$c$$ using the Pythagorean Theorem.

Accepted Answer

$$c = \sqrt{4^2$ + 2^2$} = \sqrt{20}$$

Question 10

Solve triangle $$ABC$$ using the Law of Cosines if $$a = 4$$, $$b = 3$$, $$c = 6. $$What is the measure of angle $$C (to $$the nearest degree)?

Accepted Answer

$$C = \arccos \left( \dfrac{4^2$ + 3^2$ - 6^2$}{2 \cdot 4 \cdot 3} \right) \approx 120^\circ$$

Question 11

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? (Use the Law of Cosines.)

Accepted Answer

$$\sqrt{90^2$ + 400^2$ - 2 \cdot 90 \cdot 400 \cos 45^\circ} \approx 318.2$$ feet

Question 12

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

Accepted Answer

$$C = \arccos \left( \dfrac{15^2$ + 17^2$ - 13^2$}{2 \cdot 15 \cdot 17} \right) \approx 56^\circ$$

Question 13

John wants to measure the height of a tree. He walks 100 feet from the base and looks up at an angle of $$33^\circ. $$The tree grows at an angle of $$83^\circ$$ with respect to the ground. What is the height of the tree (to the nearest foot)?

Accepted Answer

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

Question 14

Given triangle $$ABC$$ with $$\beta = 20^\circ$$, $$a = 2$$, $$c = 5$$, use the Law of Cosines to find $$b (to $$the nearest tenth).

Accepted Answer

$$b = \sqrt{2^2$ + 5^2$ - 2 \cdot 2 \cdot 5 \cos 20^\circ} \approx 3.5$$

Question 15

Find the exact value of $$	an 75^\circ - 2 \cot 15^\circ.$$

Accepted Answer

$$	an 75^\circ - 2 \cot 15^\circ = 0$$

Question 16

Given triangle $$ABC$$ with $$\beta = 40^\circ$$, $$b = 2$$, $$c = 3$$, use the Law of Sines to find $$\gamma (to $$the nearest degree).

Accepted Answer

$$\gamma = \arcsin \left( \dfrac{2 \sin 40^\circ}{3} \right) \approx 25^\circ$$

Question 17

Given triangle $$ABC$$ with $$a = 3.44$$, $$\beta = 22^\circ$$, $$c = 2.3$$, use the Law of Sines to find $$\alpha (to $$the nearest degree).

Accepted Answer

$$\alpha = \arcsin \left( \dfrac{3.44 \sin 22^\circ}{2.3} \right) \approx 34^\circ$$

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