Question

In a right triangle, the acute angles have the relationship sin(x + 12°) = cos(18° + 2x). What is the value of x? What is the measure of the smaller angle? Please use writing space to explain how you arrived at your answers, citing topics taught in this unit or show mathematical steps. (3 points)

Answer 1

To solve the equation \( \sin(x + 12^\circ) = \cos(18^\circ + 2x) \), we can use the co-function identity which states that:

\[
\sin(A) = \cos(90^\circ - A)
\]

Using this identity, we can rewrite the right side of the equation:

\[
\sin(x + 12^\circ) = \cos(18^\circ + 2x) \implies \sin(x + 12^\circ) = \sin(90^\circ - (18^\circ + 2x))
\]

Now, simplifying \( 90^\circ - (18^\circ + 2x) \):

\[
90^\circ - 18^\circ - 2x = 72^\circ - 2x
\]

Thus, we have:

\[
\sin(x + 12^\circ) = \sin(72^\circ - 2x)
\]

Now, we can set the two angles equal to each other or take into account the periodic properties of the sine function. We'll use the first case:

\[
x + 12^\circ = 72^\circ - 2x
\]

Now, let's solve for \( x \):

1. Rearrange the equation:

\[
x + 2x = 72^\circ - 12^\circ
\]

2. Combine like terms:

\[
3x = 60^\circ
\]

3. Divide by 3:

\[
x = 20^\circ
\]

Now that we have found the value of \( x \), we can determine the smaller angle in the right triangle. The triangle's angles consist of:

- One angle is \( 90^\circ \)
- One angle is \( x = 20^\circ \)
- The other angle is \( 90^\circ - x = 70^\circ \)

Since we are looking for the smaller angle:

- \( 20^\circ \) and \( 70^\circ \)

The smaller angle in the triangle is \( 20^\circ \).

In conclusion, the value of \( x \) is \( 20^\circ \), and the measure of the smaller angle in the triangle is also \( 20^\circ \).