Question

Find the measure of the indicated angle to the nearest degree

Answer 1

To find the measure of the indicated angle, we can use the Pythagorean Theorem. The two legs of the triangle have lengths 36 and 39, so the hypotenuse is equal to
`$√(36^2 + 39^2) \approx 52.47$`. We can then use the inverse sine function to find the angle whose sine is equal to
`$(36)/(52.47) \approx 0.686$`. This angle is approximately **67.38 degrees**.

To find the measure of the indicated angle in the triangle, we can use the Law of Cosines.

The Law of Cosines states that for any triangle with sides of length a, b, and c, and angle C opposite side c, the following equation holds:

`$c^2 = a^2 + b^2 - 2ab \cos(C)$`

We can rearrange this equation to solve for the angle C:

`$\cos(C) = (a^2 + b^2 - c^2)/(2ab)$`

`$C = \cos^(-1) \left( (a^2 + b^2 - c^2)/(2ab) \right)$`

In the triangle in the image, we have the following side lengths:

a = 36

b = 39

c =
`√(36^2 + 39^2) \approx 52.47$`

We want to find the measure of the angle C opposite side c.

Substituting the known side lengths into the formula for the Law of Cosines, we get:

`$\cos(C) = (36^2 + 39^2 - 52.47^2)/(2 \cdot 36 \cdot 39)$`
`$\cos(C) = (36^2 + 39^2 - 52.47^2)/(2 \cdot 36 \cdot 39)$`

`$\cos(C) \approx 0.686$`

C = \cos^{-1} (0.686)

`C \approx 67.38^\circ$`

Therefore, the measure of the indicated angle in the triangle is **67.38 degrees**.

Here is a detailed explanation of how to use the Law of Cosines to solve for the angle C:

1. Identify the known and unknown side lengths and angles. In this case, we know the side lengths a and b, and we want to find the angle C.

2. Substitute the known side lengths into the formula for the Law of Cosines.

3. Solve for the cosine of the angle C.

4. Take the inverse cosine of the cosine of the angle C to find the angle C itself.

The Law of Cosines is a powerful tool for solving triangles, and it can be used to solve for any side or angle in a triangle, given at least two other sides or angles.