Trigonometric Equations Test! Math Trivia Quiz

1) A large totem pole in the state of Washington is 100 feet tall. At a particular time of day, the totem pole casts a 249-foot long shadow. Find the measure of angle A to the nearest degree.

68

45

35

22

The measure of angle A can be found using trigonometry. The shadow of the totem pole forms a right triangle with the totem pole and the ground. The height of the totem pole is the opposite side, and the length of the shadow is the adjacent side. We can use the tangent function to find the angle. tan(A) = opposite/adjacent = 100/249. Taking the inverse tangent of this ratio, we find that A is approximately 22 degrees.

Explanation

The measure of angle A can be found using trigonometry. The shadow of the totem pole forms a right triangle with the totem pole and the ground. The height of the totem pole is the opposite side, and the length of the shadow is the adjacent side. We can use the tangent function to find the angle. tan(A) = opposite/adjacent = 100/249. Taking the inverse tangent of this ratio, we find that A is approximately 22 degrees.

2) Writing Sine in Cosine Terms and Cosine in Sine Terms Write cos 56° in terms of the sine.

The question asks to write cos 56° in terms of sine. We can use the identity sin^2θ + cos^2θ = 1 to solve this. Rearranging the equation, we get cos^2θ = 1 - sin^2θ. Substituting θ = 56°, we have cos^2(56°) = 1 - sin^2(56°). Taking the square root of both sides, we get cos(56°) = √(1 - sin^2(56°)). Therefore, the answer is sin 34, sine 34.

Explanation

The question asks to write cos 56° in terms of sine. We can use the identity sin^2θ + cos^2θ = 1 to solve this. Rearranging the equation, we get cos^2θ = 1 - sin^2θ. Substituting θ = 56°, we have cos^2(56°) = 1 - sin^2(56°). Taking the square root of both sides, we get cos(56°) = √(1 - sin^2(56°)). Therefore, the answer is sin 34, sine 34.

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3) Find the value of x. Round to the nearest degree.

67

23

83

53

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Explanation

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4) Writing Sine in Cosine Terms and Cosine in Sine Terms Write sin 38° in terms of the cosine.

The question asks to express sin 38° in terms of cosine. This can be done by using the identity sin^2θ + cos^2θ = 1. Since sin^2θ = 1 - cos^2θ, we can substitute sin^2(38°) = 1 - cos^2(38°). Taking the square root of both sides, we get sin(38°) = √(1 - cos^2(38°)). Therefore, sin 38° can be expressed in terms of the cosine as √(1 - cos^2(38°)), which is equivalent to cos 52° or cosine 52°.

Explanation

The question asks to express sin 38° in terms of cosine. This can be done by using the identity sin^2θ + cos^2θ = 1. Since sin^2θ = 1 - cos^2θ, we can substitute sin^2(38°) = 1 - cos^2(38°). Taking the square root of both sides, we get sin(38°) = √(1 - cos^2(38°)). Therefore, sin 38° can be expressed in terms of the cosine as √(1 - cos^2(38°)), which is equivalent to cos 52° or cosine 52°.

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5) Match the following

Sin

Cos

Tan

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6) A 22-foot ladder leans against a house and reaches a window that is 14 feet high. How far from the base of the house is the bottom of the ladder? Round your answer to the nearest ft.

144 feet

36 feet

17 feet

15 feet

14 feet

The ladder forms a right triangle with the ground and the wall of the house. The height of the window is the vertical side of the triangle, which is 14 feet. The ladder is the hypotenuse of the triangle, which is 22 feet. To find the distance from the base of the house to the bottom of the ladder, we need to find the length of the horizontal side of the triangle. Using the Pythagorean theorem, we can calculate this length as sqrt(22^2 - 14^2) = sqrt(484 - 196) = sqrt(288) ≈ 17 feet.

Explanation

The ladder forms a right triangle with the ground and the wall of the house. The height of the window is the vertical side of the triangle, which is 14 feet. The ladder is the hypotenuse of the triangle, which is 22 feet. To find the distance from the base of the house to the bottom of the ladder, we need to find the length of the horizontal side of the triangle. Using the Pythagorean theorem, we can calculate this length as sqrt(22^2 - 14^2) = sqrt(484 - 196) = sqrt(288) ≈ 17 feet.

7) To guard against a fall, a ladder should make an angle of 71

o to the power of blank

or less with the ground. What is the maximum height that a 25-foot ladder can reach safely? Round your answer to the nearest tenth.

The maximum height that a 25-foot ladder can reach safely can be determined by using trigonometry. The ladder forms a right triangle with the ground, where the ladder length is the hypotenuse and the height is the opposite side. The angle between the ladder and the ground is given as 71 degrees or less. By using the sine function, we can calculate the height. sin(71) = height/25. Rearranging the formula, we get height = 25 * sin(71) = 23.6 feet. Therefore, the maximum height that a 25-foot ladder can reach safely is 23.6 feet.

Explanation

The maximum height that a 25-foot ladder can reach safely can be determined by using trigonometry. The ladder forms a right triangle with the ground, where the ladder length is the hypotenuse and the height is the opposite side. The angle between the ladder and the ground is given as 71 degrees or less. By using the sine function, we can calculate the height. sin(71) = height/25. Rearranging the formula, we get height = 25 * sin(71) = 23.6 feet. Therefore, the maximum height that a 25-foot ladder can reach safely is 23.6 feet.

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8) From the top of a 160-ft high tower, at an angle of 12 degrees, an air traffic controller observes an airplane on the runway. To the nearest foot, how far from the base of the tower is the airplane?

The air traffic controller is observing the airplane from the top of a tower, which is 160 ft high. The angle of observation is given as 12 degrees. To find the distance from the base of the tower to the airplane, we can use trigonometry. We can use the tangent function, which is defined as the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tower (160 ft) and the angle is 12 degrees. By rearranging the formula, we can find the adjacent side, which represents the distance from the base of the tower to the airplane. Using a calculator, we can calculate this distance to be approximately 753 ft.

Explanation

The air traffic controller is observing the airplane from the top of a tower, which is 160 ft high. The angle of observation is given as 12 degrees. To find the distance from the base of the tower to the airplane, we can use trigonometry. We can use the tangent function, which is defined as the opposite side divided by the adjacent side. In this case, the opposite side is the height of the tower (160 ft) and the angle is 12 degrees. By rearranging the formula, we can find the adjacent side, which represents the distance from the base of the tower to the airplane. Using a calculator, we can calculate this distance to be approximately 753 ft.

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10) To find the height of a pole, a surveyor moves 144 feet away from the base of the pole and then, with a transit 4 feet tall, measures the angle of elevation to the top of the pole to be 40 degrees. To the nearest foot, what is the height of the pole?

145 ft

125 ft

135 ft

121 ft

The surveyor uses the tangent function to find the height of the pole. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the pole and the adjacent side is the distance from the surveyor to the base of the pole. The tangent of 40 degrees is equal to the height of the pole divided by 144 feet. Rearranging the equation, we get the height of the pole is equal to the tangent of 40 degrees multiplied by 144 feet. Calculating this, we find that the height of the pole is approximately 125 feet.

Explanation

The surveyor uses the tangent function to find the height of the pole. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the opposite side is the height of the pole and the adjacent side is the distance from the surveyor to the base of the pole. The tangent of 40 degrees is equal to the height of the pole divided by 144 feet. Rearranging the equation, we get the height of the pole is equal to the tangent of 40 degrees multiplied by 144 feet. Calculating this, we find that the height of the pole is approximately 125 feet.