Trigonometry: Calculating Unknown Angles

The trig ratio that should be used to calculate ϴ is sine. Sine is a trigonometric function that relates the ratio of the length of the side opposite an angle to the length of the hypotenuse in a right triangle. It is commonly used to find the value of an angle when the lengths of the sides are known. Therefore, when calculating ϴ, the sine ratio is the appropriate trigonometric function to use.

To calculate ϴ, the trig ratio that should be used is Tangent. Tangent is defined as the ratio of the length of the side opposite to the angle ϴ to the length of the adjacent side in a right triangle. It is commonly used to find the unknown side length or angle measure in a right triangle when the lengths of the other two sides are known. Therefore, to calculate ϴ, we need to use the Tangent trigonometric ratio.

The ratio that should be used to calculate ϴ is cosine. Cosine is a trigonometric function that relates the lengths of the sides of a right triangle to the angle measures. It is defined as the ratio of the adjacent side to the hypotenuse. In this context, ϴ represents an angle, and using the cosine ratio will allow us to determine the value of ϴ based on the lengths of the sides of the triangle.

To find the measure of angle θ, you can use the trigonometric ratio of cosine (cos). In a right triangle:

cos(θ) = adjacent side / hypotenuse

Given that the adjacent side is 4 units and the hypotenuse is 5 units:

cos(θ) = 4 / 5

To find θ, take the inverse cosine (arccos) of this ratio:

θ = arccos(4 / 5)

Using a calculator, you'll find that arccos(4/5) is approximately 36.87 degrees.

However, it's important to note that this angle is the complement of θ. To find θ, subtract this angle from 90 degrees (since right triangles have a 90-degree angle):

θ = 90 degrees - 36.87 degrees = 53.13 degrees

Therefore, the correct answer is option C, "53.13 degrees," which is the measure of the acute angle θ between the adjacent side and the hypotenuse.

Option A, "30 degrees," is not correct because it does not represent the angle θ in this particular triangle.

60 degrees: This is the correct answer. In a right triangle, when one acute angle measures 30 degrees, the other acute angle measures 60 degrees. This situation is known as a 30-60-90 right triangle.

The given question is asking for the value of ϴ in degrees, rounded to three significant figures. The correct answer is 33.1, which means that ϴ is equal to 33.1 degrees when rounded to three significant figures.

The given question asks for the value of ϴ in degrees, rounded to three significant figures. The answer provided is 57.8, which is already rounded to three significant figures. Therefore, there is no need to make any further adjustments or calculations.

The given question is asking for the value of ϴ, which is equal to 40.2 degrees.

The given question is asking for the value of ϴ to three significant figures. The answer provided is 46.2, which is already rounded to three significant figures. Therefore, there is no need for further calculation or rounding.

The given question asks for the value of ϴ to 3 significant figures. The answer provided is 60.6, which is already rounded to 3 significant figures. Therefore, this is the correct answer to the question.

The correct answer is 23.6 because the question asks for the value of ϴ to 3 significant figures, and the given value of 23.6 already has 3 significant figures. Therefore, there is no need to round or change the value.

The given question asks for the value of ϴ to three significant figures. The given answer, 73.9, is already rounded to three significant figures and matches the format requested by the question. Therefore, the given answer is correct.