Trigonometry Unit Test Version B

Explanation
To reach the top of a 13-foot wall using a 14-foot ladder, the ladder must be inclined at an angle of elevation. The angle of elevation is the angle between the ground and the ladder. To find this angle, we can use trigonometry. In this case, we can use the sine function, which is defined as the opposite side (height of the wall) divided by the hypotenuse (length of the ladder). So, sin(angle) = 13/14. By taking the inverse sine of both sides, we find that the angle is approximately 68.2 degrees.

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 shorter leg is half the length of the hypotenuse, and the longer leg is √3 times the length of the shorter leg. These special triangles have specific ratios between their sides, which make them useful in solving various mathematical 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 other two sides. In this case, the given lengths 13.19 and 18.85 can be the lengths of the two sides of the right triangle, and we need to find the length of the hypotenuse. By squaring both sides and rearranging the equation, we can find that the length of the hypotenuse is approximately 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 observer. In this case, the angle of depression is given as 6°. 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 adjacent side is the distance we want to find. By rearranging the equation and substituting the values, we can solve for the distance and get 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 given as 4 inches. Plugging these values into the formula, we get: angle = 16 / 4 = 4. Therefore, the central angle of the circle is 4.

Explanation
The correct answer is 2/3 π. To convert degrees to radians, we use the conversion factor of π/180. Therefore, to convert 120° to radians, we multiply 120 by π/180. Simplifying this expression gives us 2/3 π, which is the equivalent value in radians.

Explanation
To convert degrees to radians, we use the formula: radians = degrees * π/180. In this case, we have 75°, so the conversion would be 75 * π/180. Simplifying this expression gives us 5/12 π, which is the given answer.

Explanation
To convert degrees to radians, we use the formula π/180. In this case, we multiply 40° by π/180 to get the equivalent in radians. Simplifying this expression gives us 2/9 π, which is the correct answer.

Explanation
The correct answer is 240° because 4/3 π represents the radian measure of a full circle, which is 2π radians. To convert this to degrees, we multiply by 180°/π. Therefore, (4/3 π) * (180°/π) = 240°.

Explanation
The correct answer is 157.5° because when converting from radians to degrees, you multiply the given value by 180/π. In this case, 7/8 multiplied by 180/π equals 157.5°.

Explanation
To convert a fraction of π to degrees, we need to multiply the fraction by 180°. In this case, 2/5 multiplied by 180° equals 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 trigonometric function tangent. Tan(50°) = height/70. Rearranging the equation, we get height = 70 * tan(50°) = 83.4 meters.

Explanation
The man is standing at a point 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 height of the tower being the opposite side and the distance of the man from the tower being the adjacent side. We can use the tangent function to find the distance of the man from the tower. tan(60) = opposite/adjacent. Plugging in the values, we get tan(60) = 15/adjacent. Solving for adjacent, we get adjacent = 15/tan(60) = 8.66 meters, which can be rounded to 8.67 meters. Therefore, the distance of the man from the tower is 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 observer is standing 75 ft away from a tree that is 200 ft tall. To find the angle of elevation to the bird he is looking at, 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 observer to the tree (75 ft). So, the tangent of the angle of elevation is 200/75. Using a calculator, we can find that the angle of elevation is approximately 69.4°.

Explanation
The answer 10.64 is the value of x because it is the only option provided that matches the given numbers. The other options, 11.9 and 9.1, do not match any of the given numbers.

Explanation
In the given figure, we have a right-angled triangle with the height (h) and the base (d) given. We can use the trigonometric ratio tangent (tan) to find the measure of angle θ. The tangent of an angle is equal to the ratio of the opposite side to the adjacent side. In this case, tan(θ) = h/d. Substituting the given values, we get tan(θ) = 15/30 = 0.5. To find the angle, we can take the inverse tangent (arctan) of 0.5, which gives us θ = arctan(0.5) ≈ 25.67°.

Explanation
The angle that the ladder forms with the ground can be calculated using trigonometry. In this case, the ladder forms a right triangle with the wall and the ground. The length of the ladder is the hypotenuse of the triangle, and the distance of the foot of the ladder from the base of the wall is one of the legs. By using the trigonometric function tangent (opposite/adjacent), we can find the angle. The tangent of the angle is equal to the opposite side (7 feet) divided by the adjacent side (20 feet). By taking the inverse tangent (arctan) of this ratio, we find that the approximate measure of the angle is 69.5°.