Q: A tree casts a shadow of $$15.5 m. If $$the angle of elevation from the tip of the shadow to the top of the tree is $$60.4^\circ$$, how tall is the tree?

$$15.5 \tan 60.4^\circ \approx 27.3 m$$

Q: A building is $$97 ft $$tall and $$42 ft $$from point $$B. $$What is the angle of depression from the top of the building to point $$B$$?

$$\tan$$^{-1}$$\left(\frac{97}{42}\right) \approx 66.6^\circ$$

Q: Given $$\theta = 2^\circ 41' 38''$$ and $$b = 3.5000 cm$$, what is the value of $$d in $$the diagram where $$d = \frac{b}{2 \tan \theta}$$?

$$\frac{3.5000}{2 \tan 2.693888^\circ} \approx 74.43 cm$$

Q: A surveyor measures the angle of elevation to the top of a building from two points $$35 ft $$apart as $$51.8^\circ$$ and $$74.2^\circ. $$What is the height $$h of $$the building?

$$h = \frac{35 \tan 51.8^\circ \tan 74.2^\circ}{\tan 74.2^\circ - \tan 51.8^\circ} \approx 69 ft$$

Question 1

Given the right triangle with sides $$AC = 77$$, $$BC = 36$$, and hypotenuse $$AB = 85$$, what is $$\sin A$$?

Accepted Answer

$$\frac{36}{85}$$

Question 2

In the same triangle, what is $$	an B$$?

Accepted Answer

$$\frac{77}{36}$$

Question 3

For the triangle with angles $$30^\circ$$, $$60^\circ$$, and $$90^\circ$$, and sides $$1$$, $$\sqrt{3}$$, and $$2$$, what is $$\cos 30^\circ$$?

Accepted Answer

$$\frac{\sqrt{3}}{2}$$

Question 4

A ladder leans against a wall making an angle of $$28^\circ 40'$$ with the ground. If the base of the ladder is $$25.3 cm $$from the wall, what is the length of the ladder (hypotenuse)?

Accepted Answer

$$\frac{25.3}{\cos 28^\circ 40'} \approx 28.8 cm$$

Question 5

Given a right triangle with legs $$a = 44.25 cm $$and $$b = 55.87 cm$$, what is the length of the hypotenuse $$c$$?

Accepted Answer

$$c = \sqrt{44.25^2$ + 55.87^2$} \approx 71.27 cm$$

Question 6

A tree casts a shadow of $$15.5 m. If $$the angle of elevation from the tip of the shadow to the top of the tree is $$60.4^\circ$$, how tall is the tree?

Accepted Answer

$$15.5 	an 60.4^\circ \approx 27.3 m$$

Question 7

A building is $$97 ft $$tall and $$42 ft $$from point $$B. $$What is the angle of depression from the top of the building to point $$B$$?

Accepted Answer

$$	an$$^{-1}$$\left(\frac{97}{42}ight) \approx 66.6^\circ$$

Question 8

A ship sails $$8.6 km $$north-east, then turns $$53^\circ$$ and sails to point $$B. $$What is the distance $$a$$ from $$B to$$ $$C if$$ $$	riangle ABC is a $$right triangle?

Accepted Answer

$$a = 8.6 \cos 53^\circ \approx 5.2 km$$

Question 9

A ship travels $$45$$ nautical miles east, then $$54$$ nautical miles north. What is the straight-line distance from the starting point to the final position?

Accepted Answer

$$\sqrt{45^2$ + 54^2$} \approx 70$$ nautical miles

Question 10

Given $$	heta = 2^\circ 41' 38''$$ and $$b = 3.5000 cm$$, what is the value of $$d in $$the diagram where $$d = \frac{b}{2 	an 	heta}$$?

Accepted Answer

$$\frac{3.5000}{2 	an 2.693888^\circ} \approx 74.43 cm$$

Question 11

A surveyor measures the angle of elevation to the top of a building from two points $$35 ft $$apart as $$51.8^\circ$$ and $$74.2^\circ. $$What is the height $$h of $$the building?

Accepted Answer

$$h = \frac{35 	an 51.8^\circ 	an 74.2^\circ}{	an 74.2^\circ - 	an 51.8^\circ} \approx 69 ft$$

Question 12

If $$\sin 	heta = -\frac{\sqrt{3}}{2}$$ and $$	heta is $$between $$180^\circ$$ and $$360^\circ$$, what are the possible values for $$	heta$$?

Accepted Answer

$$240^\circ$$ and $$300^\circ$$

Question 13

What is the reference angle for $$294^\circ$$?

Accepted Answer

$$360^\circ - 294^\circ = 66^\circ$$

Question 14

Evaluate $$\sin 225^\circ.$$

Accepted Answer

$$-\frac{\sqrt{2}}{2}$$

Question 15

If $$	an 25^\circ \u003c 	an 23^\circ$$, is this statement true or false?

Accepted Answer

False

Question 16

Which trigonometric function decreases as the angle increases from $$0^\circ to$$ $$90^\circ$$?

Accepted Answer

Cosecant

Question 17

If $$\cot(8x) = 	an(4x + 13^\circ)$$, what is the value of $$x$$?

Accepted Answer

$$x = 17^\circ$$

Question 18

Find $$\cos 193.622^\circ to 4 $$decimal places.

Accepted Answer

$$-0.9719$$

Acute Angles and Right Triangles: Trigonometric Functions and Applications

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