Q: If $$\sin(\theta) = 0.6$$ and $$\theta is in $$the interval $$[0, 90^\circ]$$, what is $$\theta to $$the nearest degree?

$$\theta = \arcsin(0.6) \approx 37^\circ$$

Q: A building is 75 ft tall. If the angle of elevation from a point on the ground to the top of the building is $$37^\circ$$, how far is the point from the base of the building (to the nearest foot)?

$$x = \frac{75}{\tan(37^\circ)} \approx 100$$

Q: Which of the following is the correct formula for finding the angle $$A in a $$right triangle given $$\sin(A) = \frac{\text{opposite}}{\text{hypotenuse}}$$?

$$A = \arcsin\left(\frac{\text{opposite}}{\text{hypotenuse}}\right)$$

Q: Suppose the angle of elevation to the top of a tree is $$28.5^\circ$$ from a point 40 ft from the base. How tall is the tree (to the nearest foot)?

$$h = 40 \tan(28.5^\circ) \approx 22$$

Q: If $$\cos(\theta) = 0.25$$ and $$\theta is in$$ $$[0, 90^\circ]$$, what is $$\theta to $$the nearest degree?

$$\theta = \arccos(0.25) \approx 75^\circ$$

Q: Which trigonometric function would you use to find the height of a building if you know the distance from the building and the angle of elevation?

$$\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$$

Q: If $$\tan(\theta) = 2.5$$ and $$\theta is in$$ $$[0, 90^\circ]$$, what is $$\theta to $$the nearest degree?

$$\theta = \arctan(2.5) \approx 68^\circ$$

Q: A ladder leans against a wall, making a $$60^\circ$$ angle with the ground. If the foot of the ladder is 5 ft from the wall, how long is the ladder?

$$\text{Ladder length} = \frac{5}{\cos(60^\circ)} = 10$$

Question 1

Given a right triangle with legs of length $$a = 7$$ and $$b = 24$$, what is the length of the hypotenuse $$c$$?

Accepted Answer

$$c = \sqrt{7^2$ + 24^2$} = 25$$

Question 2

If $$\sin(	heta) = 0.6$$ and $$	heta is in $$the interval $$[0, 90^\circ]$$, what is $$	heta to $$the nearest degree?

Accepted Answer

$$	heta = \arcsin(0.6) \approx 37^\circ$$

Question 3

A building is 75 ft tall. If the angle of elevation from a point on the ground to the top of the building is $$37^\circ$$, how far is the point from the base of the building (to the nearest foot)?

Accepted Answer

$$x = \frac{75}{	an(37^\circ)} \approx 100$$

Question 4

Which of the following is the correct formula for finding the angle $$A in a $$right triangle given $$\sin(A) = \frac{	ext{opposite}}{	ext{hypotenuse}}$$?

Accepted Answer

$$A = \arcsin\left(\frac{	ext{opposite}}{	ext{hypotenuse}}ight)$$

Question 5

Suppose the angle of elevation to the top of a tree is $$28.5^\circ$$ from a point 40 ft from the base. How tall is the tree (to the nearest foot)?

Accepted Answer

$$h = 40 	an(28.5^\circ) \approx 22$$

Question 6

If $$\cos(	heta) = 0.25$$ and $$	heta is in$$ $$[0, 90^\circ]$$, what is $$	heta to $$the nearest degree?

Accepted Answer

$$	heta = \arccos(0.25) \approx 75^\circ$$

Question 7

A right triangle has one angle of $$35^\circ$$ and a hypotenuse of length 20. What is the length of the side adjacent to the $$35^\circ$$ angle?

Accepted Answer

$$20 \cos(35^\circ) \approx 16.4$$

Question 8

Which trigonometric function would you use to find the height of a building if you know the distance from the building and the angle of elevation?

Accepted Answer

$$	an(	heta) = \frac{	ext{opposite}}{	ext{adjacent}}$$

Question 9

If $$	an(	heta) = 2.5$$ and $$	heta is in$$ $$[0, 90^\circ]$$, what is $$	heta to $$the nearest degree?

Accepted Answer

$$	heta = \arctan(2.5) \approx 68^\circ$$

Question 10

A ladder leans against a wall, making a $$60^\circ$$ angle with the ground. If the foot of the ladder is 5 ft from the wall, how long is the ladder?

Accepted Answer

$$	ext{Ladder length} = \frac{5}{\cos(60^\circ)} = 10$$

Question 11

If $$\sin(	heta) = 0.8$$ and $$	heta is in$$ $$[0, 90^\circ]$$, what is $$	heta to $$the nearest degree?

Accepted Answer

$$	heta = \arcsin(0.8) \approx 53^\circ$$

Question 12

A surveyor measures the angle of elevation to the top of a building as $$40^\circ$$ from a distance of 60 ft. What is the height of the building to the nearest foot?

Accepted Answer

$$h = 60 	an(40^\circ) \approx 50$$

Question 13

Which of the following is the correct definition of the tangent function for a right triangle?

Accepted Answer

$$	an(	heta) = \frac{	ext{opposite}}{	ext{adjacent}}$$

Question 14

If $$\cos(	heta) = 0.5$$ and $$	heta is in$$ $$[0, 90^\circ]$$, what is $$	heta$$?

Accepted Answer

$$	heta = 60^\circ$$

Question 15

A right triangle has legs of length 9 and 12. What is the measure of the angle opposite the side of length 9 (to the nearest degree)?

Accepted Answer

$$	heta = \arctan\left(\frac{9}{12}ight) \approx 37^\circ$$

Question 16

If $$	an(	heta) = 1$$, what is $$	heta in$$ $$[0, 90^\circ]$$?

Accepted Answer

$$	heta = 45^\circ$$

Angles, Right Triangles, and Trigonometric Function Values: Study Notes

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