Free Pre-Calculus Trigonometry Questions and Answers — Questions and Answers

Question 1: Find the length of AC for the given triangle if tan(x) = 40/9.

50
40
49
41 (Correct answer)

Correct answer: 41

Given tan(x) = 40/9, this means the opposite side (BC) is 40 and the adjacent side (AB) is 9 in a right triangle. To find the hypotenuse (AC), use the Pythagorean theorem: AC^2 = AB^2 + BC^2. So, AC^2 = 9^2 + 40^2 = 81 + 1600 = 1681. Taking the square root of 1681 gives AC = 41.

Question 2: Which way do you measure a positive angle?

Clockwise
Counterclockwise (Correct answer)
Up
Down

Correct answer: Counterclockwise

In trigonometry and mathematics, positive angles are conventionally measured by rotating the terminal side counterclockwise from the positive x-axis. This standard convention helps maintain consistency in angle definitions and trigonometric functions across different quadrants. Clockwise rotation, conversely, defines negative angles.

Question 3: Determine the degree size of one complete revolution.

180
90
360 (Correct answer)
270

Correct answer: 360

One complete revolution around a circle is defined as 360 degrees. This measurement system divides a circle into 360 equal parts, with each part representing one degree. This convention is widely used in geometry, navigation, and many engineering applications.

Question 4: Which measurement system is most commonly used to express the size of an angle?

Radians
Revolutions
Degrees (Correct answer)
Gradians

Correct answer: Degrees

The degree system is the most commonly used unit for expressing the size of an angle in everyday contexts and many practical applications. While radians are fundamental in higher mathematics and calculus, degrees provide a more intuitive and easily visualized measure for general use. A full circle is 360 degrees, a half circle is 180 degrees, and a quarter circle is 90 degrees.

Question 5: What is the approximate value of π (pi) ?

2.14
3.14 (Correct answer)
2.16
3.15

Correct answer: 3.14

Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. Its approximate value is 3.14159, but for most calculations, 3.14 is a sufficiently accurate approximation. This value is fundamental in geometry and trigonometry, appearing in formulas for areas, volumes, and arc lengths.

Question 6: What is an angle's radian value for a single revolution?

π/2
π
2π (Correct answer)
π/4

Correct answer: 2π

In the radian system, one complete revolution around a circle is equal to 2π radians. This is derived from the definition of a radian as the angle subtended by an arc whose length is equal to the radius of the circle. Since the circumference of a circle is 2πr, a full circle encompasses 2π radians.

Question 7: Find the cosine of a if the sides a and b of a right triangle are 8 and 6, respectively.

3/5 (Correct answer)
2/5
1/5
5/3

Correct answer: 3/5

Given sides a=8 and b=6, these are the legs of a right triangle. First, find the hypotenuse (c) using the Pythagorean theorem: c^2 = a^2 + b^2 = 8^2 + 6^2 = 64 + 36 = 100, so c = 10. The cosine of angle 'a' (assuming 'a' is the angle opposite side 'a') is defined as the adjacent side divided by the hypotenuse. The side adjacent to angle 'a' is 'b' (6), and the hypotenuse is 'c' (10). Therefore, cos(a) = b/c = 6/10 = 3/5.

Question 8: If a is 3 and c is 5, find the tangent of a right triangle.

5/3
3/4 (Correct answer)
1/4
4/3

Correct answer: 3/4

Given side a=3 and hypotenuse c=5. First, find side b (adjacent to angle 'a') using the Pythagorean theorem: b^2 = c^2 - a^2 = 5^2 - 3^2 = 25 - 9 = 16, so b = 4. The tangent of angle 'a' is defined as the opposite side divided by the adjacent side. The side opposite angle 'a' is 'a' (3), and the side adjacent to angle 'a' is 'b' (4). Therefore, tan(a) = a/b = 3/4.

Question 9: Turn 360 degrees into radians.

π/3
4π
π/2
2π (Correct answer)

Correct answer: 2π

To convert degrees to radians, use the conversion factor (π radians / 180 degrees). For 360 degrees, the calculation is 360 * (π/180). This simplifies to 2π radians. This conversion highlights that a full circle, or one complete revolution, is equivalent to 2π radians.

Question 10: Use the Pythagorean theorem to find the length of the hypotenuse, c.

244
2√61 (Correct answer)
2√16
2√122

Correct answer: 2√61

The Pythagorean theorem states that in a right triangle, the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b), i.e., a² + b² = c². If we assume the legs are 10 and 12 (a common example for such problems), then c² = 10² + 12² = 100 + 144 = 244. Therefore, c = √244, which simplifies to 2√61.

Question 11: If you stand 47 feet from a tree's base, the angle of elevation between you and the treetop is 35 degrees. Determine the nearest foot measurement of the tree's height.

33 (Correct answer)
38
67
27

Correct answer: 33

This is a right triangle problem where the distance from the tree (adjacent side) is 47 feet and the angle of elevation is 35 degrees. We need to find the tree's height (opposite side). The tangent function relates the opposite and adjacent sides: tan(angle) = opposite/adjacent. So, tan(35°) = height / 47. Solving for height: height = 47 * tan(35°) ≈ 47 * 0.7002 ≈ 32.9094. Rounded to the nearest foot, the height is 33 feet.

Question 12: You decide to measure the height of the bungee jumping platform while waiting for your sister to complete her jump. There is an elevation angle of 62 degrees between where you are standing and the top of the platform, which is 200 feet away. Can you tell me how many feet the platform is?

298.05
449.71
144.5
376.15 (Correct answer)

Correct answer: 376.15

This scenario forms a right triangle where the distance from the platform (adjacent side) is 200 feet and the angle of elevation is 62 degrees. We need to find the platform's height (opposite side). Using the tangent function: tan(angle) = opposite/adjacent. So, tan(62°) = height / 200. Solving for height: height = 200 * tan(62°) ≈ 200 * 1.8807 ≈ 376.14. Rounded to two decimal places, the height is 376.15 feet.

Question 13: A flagpole's shadow is 10 feet long when the sun is at an elevation of 32 degrees. What is the flagpole's height, in feet?

6.25 (Correct answer)
5.75
7.5
6.5

Correct answer: 6.25

This problem involves a right triangle where the shadow length (adjacent side) is 10 feet and the angle of elevation of the sun is 32 degrees. We need to find the flagpole's height (opposite side). The tangent function is appropriate here: tan(angle) = opposite/adjacent. So, tan(32°) = height / 10. Solving for height: height = 10 * tan(32°) ≈ 10 * 0.6248 ≈ 6.248. Rounded to two decimal places, the height is 6.25 feet.

Question 14: A 125-foot-long ladder is resting at a 70-degree angle against the wall of a house. How high, in feet, can the ladder go up the house's side?

184.51
117.46 (Correct answer)
166.44
109.69

Correct answer: 117.46

The ladder forms the hypotenuse (125 feet) of a right triangle, and the angle it makes with the ground is 70 degrees. We need to find how high the ladder reaches on the wall (opposite side). The sine function relates the opposite side and the hypotenuse: sin(angle) = opposite/hypotenuse. So, sin(70°) = height / 125. Solving for height: height = 125 * sin(70°) ≈ 125 * 0.9397 ≈ 117.4625. Rounded to two decimal places, the height is 117.46 feet.

Question 15: What is the length of side BC in the right triangle ABC, whose angle A is 90 degrees, whose side AB is 15, and whose side AC is 36?

40
33
39 (Correct answer)
This triangle cannot exist.

Correct answer: 39

Given a right triangle ABC with angle A = 90 degrees, side AB = 15, and side AC = 36. We need to find the length of side BC, which is the hypotenuse since it's opposite the right angle A. Using the Pythagorean theorem: BC^2 = AB^2 + AC^2. So, BC^2 = 15^2 + 36^2 = 225 + 1296 = 1521. Taking the square root of 1521 gives BC = 39.

Question 16: Two posts placed 25 meters apart host a zip line. The zip line is slanted at an angle of 10 degrees from horizontal. Find out how long the zip line is.

24.6m
144.0m
25.4m (Correct answer)
141.8m

Correct answer: 25.4m

The two posts are 25 meters apart, which represents the adjacent side of a right triangle formed by the zip line, the horizontal distance, and the height difference. The zip line is the hypotenuse, and the angle it makes with the horizontal is 10 degrees. We use the cosine function, which relates the adjacent side and the hypotenuse: cos(angle) = adjacent/hypotenuse. So, cos(10°) = 25 / zip line length. Solving for zip line length: zip line length = 25 / cos(10°) ≈ 25 / 0.9848 ≈ 25.3858. Rounded to one decimal place, the zip line is 25.4 meters long.

Free Pre-Calculus Trigonometry Questions and Answers

Find the length of AC for the given triangle if tan(x) = 40/9.