Trigonometry Unit Test (Honors)

Explanation
To reach the top of the 13-foot wall, a 14-foot ladder is used, which means the ladder will extend beyond the top of the wall. The angle of elevation represents the angle between the ground and the ladder. In order to reach the top of the wall, the ladder needs to be inclined at an angle that allows it to extend beyond the wall. The correct answer of 68.2 degrees indicates that the ladder should be inclined at this angle to reach the top of the wall.

Explanation
The two special triangles mentioned in the answer are the 45-45-90 triangle and the 30-60-90 triangle. In a 45-45-90 triangle, the two legs are congruent and the hypotenuse is √2 times the length of each leg. In a 30-60-90 triangle, the lengths of the sides are in the ratio 1:√3:2, where the shortest side is opposite the 30-degree angle, the longest side is opposite the 90-degree angle, and the remaining side is opposite the 60-degree angle. These special triangles have specific properties that make them useful in solving geometric problems.

Explanation
The length of the opposite side can be found using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In this case, if we assume that the given lengths are the lengths of the two sides adjacent to the right angle, then the length of the opposite side can be found by taking the square root of the difference between the square of the hypotenuse and the square of one of the adjacent sides. In this case, the length of the opposite side is 16.10.

Explanation
The angle of depression is the angle formed between a horizontal line and the line of sight from an observer to a point below the horizontal line. In this case, the angle of depression is given as 6°. The height of the lighthouse is given as 100 meters. To find the distance of the boat from the lighthouse, we can use the tangent function. 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 lighthouse (100 meters) and the angle is 6°. By rearranging the formula, we can solve for the adjacent side, which represents the distance of the boat from the lighthouse. Using the formula, we find that the distance is approximately 952.4 meters.

Explanation
The central angle of a circle can be found using the formula: angle = arc length / radius. In this case, the arc length is given as 16 inches and the radius is 4 inches. Therefore, the central angle is 16/4 = 4.

Explanation
The angle measurement of 120° can be converted to radians by multiplying it by π/180. This gives us 120° * π/180 = 2/3 π. Therefore, the correct answer is 2/3 π.

Explanation
To convert degrees to radians, we use the formula: radians = degrees * π/180. In this case, 75° is being converted to radians. Using the formula, we get: radians = 75 * π/180. Simplifying this expression, we get: radians = 5π/12. Therefore, the correct answer is 5/12 π.

Explanation
The question is asking for the value of 40° in radians. To convert degrees to radians, we use the formula π/180. Therefore, 40° in radians is (40° * π/180) = 2/9 π.

Explanation
The question is asking for the value of 4/3 π in degrees. To convert from radians to degrees, we multiply the radian measure by 180/π. Therefore, 4/3 π * 180/π = 240°. So, the correct answer is 240°.

Explanation
The correct answer is 157.5° because 7/8 of a full circle is equivalent to 315°, and since there are 360° in a full circle, we can subtract 315° from 360° to find the remaining angle measure, which is 45°. Dividing this angle by 2 gives us 22.5°, and adding it to the original angle of 135° gives us a final answer of 157.5°.

Explanation
To convert radians to degrees, we need to multiply the given value by 180/π. In this case, 2/5 π is equivalent to (2/5) * (180/π) degrees. Simplifying this, we get (2/5) * (180/π) = 72°. Therefore, the correct answer is 72°.

Explanation
The photographer is standing 70 meters away from the building and the angle of elevation is 50°. To find the height of the building, we can use the tangent function. 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 building and the adjacent side is the distance from the photographer to the building. So, we have tan(50°) = height/70. Rearranging the equation, we get height = 70 * tan(50°) ≈ 83.4 meters. Therefore, the height of the building is approximately 83.4 meters.

Explanation
The man is standing on the ground and notices that the top of the tower makes an angle of 60 degrees with the ground. This forms a right triangle with the tower as the height (h) and the distance of the man from the tower as the base. The height of the tower is given as 15 meters. Using trigonometry, we can use the tangent function to find the distance of the man from the tower. tan(60 degrees) = h/base. Rearranging the equation, we get base = h/tan(60 degrees) = 15/tan(60 degrees) = 8.67 meters.

Explanation
The angle of elevation is the angle between the line of sight from the observer to the object and the horizontal plane. In this case, the man is standing 75 ft away from the tree and looking at a bird on top of the tree. To find the angle of elevation, we can use the tangent function. 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 tree (200 ft) and the adjacent side is the distance from the man to the tree (75 ft). Therefore, the angle of elevation can be calculated as the arctan(200/75) which is approximately 69.4°.

Explanation
The given numbers 30.48, 140.55, and 173.21 are most likely measurements in centimeters. Since the question asks to find the height, it can be inferred that the numbers represent the heights of different objects. Among these numbers, 173.21 is the highest, indicating that it is the tallest object or the height being referred to in the question.

Explanation
In the given figure, we have a right triangle with one angle of 30° and the opposite side length of 15 m. We are required to find the measure of angle θ. Using trigonometric ratios, we can use the sine function to find θ. The sine of θ is equal to the ratio of the opposite side (15 m) to the hypotenuse (30 m). So, sin(θ) = 15/30 = 0.5. Taking the inverse sine of 0.5, we find that θ is approximately 25.67°.

Explanation
The approximate measure of the angle the ladder forms with the ground can be found using trigonometry. In this case, we can use the tangent function, which is defined as the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle. The length of the side opposite the angle is the height of the wall, which is 20 feet, and the length of the side adjacent to the angle is the distance from the base of the wall to the foot of the ladder, which is 7 feet. Therefore, the tangent of the angle is 20/7. Taking the inverse tangent of this value, we get approximately 69.5°.

Explanation
In a right triangle, the opposite side is the side that is opposite to the given angle. In this case, the opposite side is 5. Since the angle is 45 degrees, it means that the triangle is an isosceles right triangle. In an isosceles right triangle, the two legs are congruent. Therefore, the length of the other leg (side x) is also 5.

Explanation
The given answer, 22.62, is the correct angle. It is likely that this angle was calculated using trigonometry, possibly using the sine or cosine function. Without further context or information, it is difficult to provide a more specific explanation.