Q: Which of the following is the correct definition of $$\tan \theta in a $$right triangle?

$$\dfrac{\text{opposite}}{\text{adjacent}}$$

Q: A 20-foot shadow is cast by a tree when the angle of elevation of the sun is $$30^\circ. $$How tall is the tree?

$$20 \cdot \tan 30^\circ \approx 11.55 ft$$

Q: A pilot is flying at an altitude of 5000 ft and observes an airport at an angle of depression of $$27^\circ. $$What is the horizontal distance to the airport?

$$\dfrac{5000}{\tan 27^\circ}$$

Question 1

Given a right triangle where one angle is $$45^\circ$$ and the legs are both of length 1, what is the value of $$\sin 45^\circ$$?

Accepted Answer

$$\dfrac{\sqrt{2}}{2}$$

Question 2

A ladder leans against a wall making a $$60^\circ$$ angle with the ground. If the ladder is 10 ft long, how high up the wall does it reach? (Assume the wall is perpendicular to the ground.)

Accepted Answer

$$10 \cdot \sin 60^\circ = 5\sqrt{3} ft$$

Question 3

Which of the following is the correct definition of $$	an 	heta in a $$right triangle?

Accepted Answer

$$\dfrac{	ext{opposite}}{	ext{adjacent}}$$

Question 4

A surveyor measures the angle of elevation to the top of a building as $$27^\circ$$ from a point 50 meters from the base. How tall is the building?

Accepted Answer

$$50 \cdot 	an 27^\circ$$

Question 5

Which of the following is a common calculator error when evaluating trigonometric functions?

Accepted Answer

Using the calculator in the wrong angle mode (degree/radian)

Question 6

Given a $$30^\circ$$-$$60^\circ$$-$$90^\circ$$ triangle with hypotenuse 2, what is the length of the side opposite the $$30^\circ$$ angle?

Accepted Answer

$$1$$

Question 7

If $$\sin 	heta = 0.5$$ and $$	heta is an $$acute angle, what is $$	heta$$?

Accepted Answer

$$30^\circ$$

Question 8

A right triangle has one leg of length 6 and an adjacent angle of $$27^\circ. $$What is the length of the other leg (opposite the $$27^\circ$$ angle)?

Accepted Answer

$$6 \cdot 	an 27^\circ$$

Question 9

Which of the following is the value of $$\cos 60^\circ$$?

Accepted Answer

$$\dfrac{1}{2}$$

Question 10

A ramp rises 3 feet over a horizontal distance of 12 feet. What is the angle of elevation of the ramp, to the nearest degree?

Accepted Answer

$$\arctan \left( \dfrac{3}{12} \right) \approx 14^\circ$$

Question 11

Which of the following is the correct value for $$	an 45^\circ$$?

Accepted Answer

$$1$$

Question 12

If $$\cos 	heta = \dfrac{1}{2}$$ and $$	heta is an $$acute angle, what is $$	heta$$?

Accepted Answer

$$60^\circ$$

Question 13

A 20-foot shadow is cast by a tree when the angle of elevation of the sun is $$30^\circ. $$How tall is the tree?

Accepted Answer

$$20 \cdot 	an 30^\circ \approx 11.55 ft$$

Question 14

Which of the following is the correct value for $$\sin 60^\circ$$?

Accepted Answer

$$\dfrac{\sqrt{3}}{2}$$

Question 15

A right triangle has sides of length 1, 1, and $$\sqrt{2}. $$What are the measures of its non-right angles?

Accepted Answer

$$45^\circ$$ and $$45^\circ$$

Question 16

If $$	an 	heta = \sqrt{3}$$ and $$	heta is an $$acute angle, what is $$	heta$$?

Accepted Answer

$$60^\circ$$

Question 17

A construction worker needs to find the length of a roof rafter that rises at a $$27^\circ$$ angle from the horizontal and spans a horizontal distance of 12 feet. What is the length of the rafter?

Accepted Answer

$$\dfrac{12}{\cos 27^\circ}$$

Question 18

Which of the following is the reciprocal of $$\sin 	heta$$?

Accepted Answer

$$\csc 	heta$$

Question 19

A pilot is flying at an altitude of 5000 ft and observes an airport at an angle of depression of $$27^\circ. $$What is the horizontal distance to the airport?

Accepted Answer

$$\dfrac{5000}{	an 27^\circ}$$

Trigonometry of Acute Angles: Right Triangle Trigonometry and Applications

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