Trigonometric Functions Evaluation Quiz

1. Cat problem: Your cat is stuck on a tree branch 6.5m above the ground. Your ladder is only 6.7m long. If you place the ladder's tip on the tree branch, what angle will the ladder make with the ground?

75°

62°

76°

72°

The ladder will make an angle of 76° with the ground. This can be determined by using trigonometry. The ladder, the height of the tree branch, and the distance from the base of the ladder to the tree form a right triangle. The length of the ladder is the hypotenuse of the triangle, the height of the tree branch is the opposite side, and the distance from the base of the ladder to the tree is the adjacent side. By using the inverse sine function, we can find the angle. The inverse sine of (6.5/6.7) is approximately 76°.

Explanation

The ladder will make an angle of 76° with the ground. This can be determined by using trigonometry. The ladder, the height of the tree branch, and the distance from the base of the ladder to the tree form a right triangle. The length of the ladder is the hypotenuse of the triangle, the height of the tree branch is the opposite side, and the distance from the base of the ladder to the tree is the adjacent side. By using the inverse sine function, we can find the angle. The inverse sine of (6.5/6.7) is approximately 76°.

2. Grand piano problem: The lid on a grand piano is held open by a prop 28 In long. The base of the prop is 55 in from the lid's hinge. At what angle will the lid open when the prop is placed so that it makes a right angle with the lid?

52°

24°

42°

31°

The correct answer is 31°. To find the angle at which the lid will open, we can use trigonometry. The prop and the lid form a right triangle, with the prop as the hypotenuse and the base of the prop as one of the legs. We can use the trigonometric function tangent to find the angle. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side is the height of the prop (28 in) and the adjacent side is the distance from the hinge to the base of the prop (55 in). Taking the inverse tangent (arctan) of this ratio gives us the angle, which is approximately 31°.

Explanation

The correct answer is 31°. To find the angle at which the lid will open, we can use trigonometry. The prop and the lid form a right triangle, with the prop as the hypotenuse and the base of the prop as one of the legs. We can use the trigonometric function tangent to find the angle. The tangent of an angle is equal to the length of the opposite side divided by the length of the adjacent side. In this case, the opposite side is the height of the prop (28 in) and the adjacent side is the distance from the hinge to the base of the prop (55 in). Taking the inverse tangent (arctan) of this ratio gives us the angle, which is approximately 31°.

3. Submarine problem: A submarine at the surface of the ocean makes an emergency div, making an angle of 21° with the surface. If the submarine travels 300 m, how deep will it be?

107.510 m

115.246 m

205.246 m

210.47 m

The correct answer is 107.510 m. When the submarine makes an emergency dive at an angle of 21° with the surface, it forms a right triangle with the surface of the ocean. The distance traveled by the submarine represents the adjacent side of the triangle, and the depth of the submarine represents the opposite side. We can use the trigonometric function tangent to find the depth of the submarine. By using the formula tan(angle) = opposite/adjacent, we can rearrange the formula to solve for the opposite side, which is the depth. Plugging in the values, we get tan(21°) = depth/300. Solving for the depth gives us a value of approximately 107.510 m.

Explanation

The correct answer is 107.510 m. When the submarine makes an emergency dive at an angle of 21° with the surface, it forms a right triangle with the surface of the ocean. The distance traveled by the submarine represents the adjacent side of the triangle, and the depth of the submarine represents the opposite side. We can use the trigonometric function tangent to find the depth of the submarine. By using the formula tan(angle) = opposite/adjacent, we can rearrange the formula to solve for the opposite side, which is the depth. Plugging in the values, we get tan(21°) = depth/300. Solving for the depth gives us a value of approximately 107.510 m.

4. Street inclination problem: Someone wants to know the inclination angle of a street. He notices that the bricks in a fence are horizontal with respect to the street and that the horizontal distance from a point is 35 cm. From there, downward, the brick measures 6 cm of vertical distance toward the street. What is the inclination angle?

12° 13’12”

9°43´39.28”

80°16’20”

10°19´20"

None

The correct answer of 9°43´39.28” can be explained by using the concept of trigonometry. The problem provides the horizontal distance (35 cm) and the vertical distance (6 cm) of the brick from a point. By using the tangent function, we can calculate the angle of inclination. The tangent of an angle is equal to the ratio of the opposite side (vertical distance) to the adjacent side (horizontal distance). In this case, the tangent of the angle is 6 cm / 35 cm. By taking the inverse tangent (arctan) of this ratio, we can find the angle of inclination, which is approximately 9°43´39.28”.

Explanation

The correct answer of 9°43´39.28” can be explained by using the concept of trigonometry. The problem provides the horizontal distance (35 cm) and the vertical distance (6 cm) of the brick from a point. By using the tangent function, we can calculate the angle of inclination. The tangent of an angle is equal to the ratio of the opposite side (vertical distance) to the adjacent side (horizontal distance). In this case, the tangent of the angle is 6 cm / 35 cm. By taking the inverse tangent (arctan) of this ratio, we can find the angle of inclination, which is approximately 9°43´39.28”.

5. Radiotherapy problem: A gamma-ray tube is used to treat a tumor located 5.7 cm under the skin of the patient. In order not to damage an organ on top of the tumor, the technician moves the tube 8.3 cm. to one side. What would be the angle of the tube for the gamma rays to affect the tumor?

34°28’45”

55°31’14”

34°11’19”

51°35’24”

35°23’44”

The angle of the tube for the gamma rays to affect the tumor can be calculated using trigonometry. Since the tumor is located 5.7 cm under the skin and the technician moves the tube 8.3 cm to one side, we can consider a right triangle with the hypotenuse being the distance from the tube to the tumor (8.3 cm) and the opposite side being the depth of the tumor (5.7 cm). The angle can be found using the inverse tangent function, which gives us an angle of 34°28’45”.

Explanation

The angle of the tube for the gamma rays to affect the tumor can be calculated using trigonometry. Since the tumor is located 5.7 cm under the skin and the technician moves the tube 8.3 cm to one side, we can consider a right triangle with the hypotenuse being the distance from the tube to the tumor (8.3 cm) and the opposite side being the depth of the tumor (5.7 cm). The angle can be found using the inverse tangent function, which gives us an angle of 34°28’45”.

6. Flag rope problem: If you need to buy a rope for the flag pole and you notice that the shadow cast by the flagpole is 11.6m. While the elevation angle of the sun is 35° 40.’ What is the length of the rope you must buy?

16.65m

32.32m

17.28m

34.23m

15.38m

Given that the shadow cast by the flagpole is 11.6m and the elevation angle of the sun is 35° 40', we can use trigonometry to find the length of the rope. The shadow cast by the flagpole represents the adjacent side of a right triangle, while the height of the flagpole represents the opposite side. We can use the tangent function to find the length of the rope, which is the hypotenuse of the triangle. By using the formula tan(angle) = opposite/adjacent, we can calculate the length of the rope to be approximately 16.65m.

Explanation

Given that the shadow cast by the flagpole is 11.6m and the elevation angle of the sun is 35° 40', we can use trigonometry to find the length of the rope. The shadow cast by the flagpole represents the adjacent side of a right triangle, while the height of the flagpole represents the opposite side. We can use the tangent function to find the length of the rope, which is the hypotenuse of the triangle. By using the formula tan(angle) = opposite/adjacent, we can calculate the length of the rope to be approximately 16.65m.

7. Radiotherapy problem: A gamma-ray tube is used to treat a tumor located 5.7 cm under the skin of the patient. In order not to damage an organ on top of the tumor, the technician moves the tube 8.3 cm. to one side.4 How long would the ray travel through the skin?

11.45cm

16.10cm

10.06cm

15.23cm

12.06cm

The ray would travel through the skin for a distance of 10.06 cm. This can be calculated by using the Pythagorean theorem to find the hypotenuse of a right triangle. The distance under the skin (5.7 cm) and the distance moved to one side (8.3 cm) form the two legs of the triangle. By squaring each leg, adding them together, and taking the square root of the sum, we can find the length of the hypotenuse, which represents the distance the ray travels through the skin.

Explanation

The ray would travel through the skin for a distance of 10.06 cm. This can be calculated by using the Pythagorean theorem to find the hypotenuse of a right triangle. The distance under the skin (5.7 cm) and the distance moved to one side (8.3 cm) form the two legs of the triangle. By squaring each leg, adding them together, and taking the square root of the sum, we can find the length of the hypotenuse, which represents the distance the ray travels through the skin.

8. Ladder problem: You lean a ladder that is 6.7m long against the wall. It makes an angle of 63° with the level ground. How high is the top of the ladder?

3.724 m

4.348 m

4.514 m

5.970 m

The height of the top of the ladder can be calculated using trigonometry. In this case, we can use the sine function to find the height. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the height of the ladder and the hypotenuse is the length of the ladder. So, we can set up the equation sin(63°) = height/6.7m. Solving for height, we find that the height of the top of the ladder is approximately 5.970 m.

Explanation

The height of the top of the ladder can be calculated using trigonometry. In this case, we can use the sine function to find the height. The sine of an angle is equal to the opposite side divided by the hypotenuse. In this case, the opposite side is the height of the ladder and the hypotenuse is the length of the ladder. So, we can set up the equation sin(63°) = height/6.7m. Solving for height, we find that the height of the top of the ladder is approximately 5.970 m.

9. Lighthouse problem: An observer watches from the top of a 60m height lighthouse a boat on the water with a depression angle of 12°42.’ What is the distance from the boat to the tower?

58.53m

13.19m

13.52m

57.27m

13.89m

The correct answer is 58.53m. The observer on top of the lighthouse is looking down at the boat on the water, creating a depression angle. To find the distance from the boat to the tower, we can use trigonometry. The tangent of the depression angle is equal to the height of the lighthouse divided by the distance from the boat to the tower. By rearranging the equation, we can solve for the distance, which is approximately 58.53m.

Explanation

The correct answer is 58.53m. The observer on top of the lighthouse is looking down at the boat on the water, creating a depression angle. To find the distance from the boat to the tower, we can use trigonometry. The tangent of the depression angle is equal to the height of the lighthouse divided by the distance from the boat to the tower. By rearranging the equation, we can solve for the distance, which is approximately 58.53m.

10. Pyramid problem: The Great Pyramid of Cheops in Egypt has a square base 230 m on each side. The faces of the pyramid make an angle of 52° with the horizontal. How tall is the pyramid?

130.156 m

147.193 m

153.132 m

167.275 m

The height of the pyramid can be determined using trigonometry. Since the pyramid has a square base, the height forms a right angle with the base. The given angle of 52° is the angle between the height and the horizontal. By using the tangent function, we can calculate the height. The formula is height = base * tan(angle). Plugging in the values, we get height = 230 * tan(52°) = 147.193 m.

Explanation

The height of the pyramid can be determined using trigonometry. Since the pyramid has a square base, the height forms a right angle with the base. The given angle of 52° is the angle between the height and the horizontal. By using the tangent function, we can calculate the height. The formula is height = base * tan(angle). Plugging in the values, we get height = 230 * tan(52°) = 147.193 m.