Question
In a right triangle, the acute angles have the relationship sin(x + 12°) = cos (18° + 2x)
1. What is the value of x?
2. What is the measure of the smaller angle?
3. Use 1 sentence (in a 8th grader response) to explain how you arrived at your answers.
1. What is the value of x?
2. What is the measure of the smaller angle?
3. Use 1 sentence (in a 8th grader response) to explain how you arrived at your answers.
Answers
GPT-4o mini
To solve the equation \( \sin(x + 12^\circ) = \cos(18^\circ + 2x) \), we can use the co-function identity \( \sin(A) = \cos(90^\circ - A) \). This gives us:
\[
\sin(x + 12^\circ) = \cos(18^\circ + 2x) \implies x + 12^\circ = 90^\circ - (18^\circ + 2x)
\]
Simplifying this, we get:
\[
x + 12^\circ = 90^\circ - 18^\circ - 2x
\]
Combining like terms:
\[
x + 2x + 12^\circ = 72^\circ
\]
\[
3x + 12^\circ = 72^\circ
\]
\[
3x = 72^\circ - 12^\circ = 60^\circ
\]
\[
x = 20^\circ
\]
The two acute angles in the triangle are \( x \) and \( 90^\circ - (x + 12^\circ) \):
\[
90^\circ - (20^\circ + 12^\circ) = 90^\circ - 32^\circ = 58^\circ
\]
Thus, \( x = 20^\circ \) and the smaller angle is \( 20^\circ \).
1. The value of \( x \) is \( 20^\circ \).
2. The measure of the smaller angle is \( 20^\circ \).
3. I used the co-function identity for sine and cosine, set the angles equal, simplified the equation, and solved for \( x \).
\[
\sin(x + 12^\circ) = \cos(18^\circ + 2x) \implies x + 12^\circ = 90^\circ - (18^\circ + 2x)
\]
Simplifying this, we get:
\[
x + 12^\circ = 90^\circ - 18^\circ - 2x
\]
Combining like terms:
\[
x + 2x + 12^\circ = 72^\circ
\]
\[
3x + 12^\circ = 72^\circ
\]
\[
3x = 72^\circ - 12^\circ = 60^\circ
\]
\[
x = 20^\circ
\]
The two acute angles in the triangle are \( x \) and \( 90^\circ - (x + 12^\circ) \):
\[
90^\circ - (20^\circ + 12^\circ) = 90^\circ - 32^\circ = 58^\circ
\]
Thus, \( x = 20^\circ \) and the smaller angle is \( 20^\circ \).
1. The value of \( x \) is \( 20^\circ \).
2. The measure of the smaller angle is \( 20^\circ \).
3. I used the co-function identity for sine and cosine, set the angles equal, simplified the equation, and solved for \( x \).