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 7th grader response) to explain how you arrived at your answers.
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 7th grader response) to explain how you arrived at your answers.
Answers
GPT-4o mini
To solve the equation \( \sin(x + 12°) = \cos(18° + 2x) \):
1. Using the trigonometric identity \( \cos(θ) = \sin(90° - θ) \), we can rewrite the equation:
\[
\sin(x + 12°) = \sin(90° - (18° + 2x))
\]
Simplifying the right side gives:
\[
\sin(x + 12°) = \sin(72° - 2x)
\]
Now we can set the angles equal to each other or equal to their supplements:
\[
x + 12° = 72° - 2x \quad \text{or} \quad x + 12° = 180° - (72° - 2x)
\]
Solving the first equation:
\[
x + 2x = 72° - 12°
\]
\[
3x = 60°
\]
\[
x = 20°
\]
We will check the second equation:
\[
x + 12° = 108° + 2x
\]
\[
-x = 108° - 12°
\]
\[
-x = 96°
\]
\[
x = -96° \quad (\text{not valid as angles cannot be negative})
\]
Thus, the only solution is:
**x = 20°**
2. The smaller acute angle is \( x = 20° \), and the other angle can be found as follows:
Since the angles in a triangle must sum to \( 90° \) for a right triangle, the other acute angle \( y \) is:
\[
y = 90° - x = 90° - 20° = 70°
\]
Among these, \( 20° \) is the smallest angle.
3. I found the value of \( x \) by setting the two sine expressions equal to each other, solved for \( x \), and then determined the smaller angle in the triangle by subtracting \( x \) from \( 90° \).
1. Using the trigonometric identity \( \cos(θ) = \sin(90° - θ) \), we can rewrite the equation:
\[
\sin(x + 12°) = \sin(90° - (18° + 2x))
\]
Simplifying the right side gives:
\[
\sin(x + 12°) = \sin(72° - 2x)
\]
Now we can set the angles equal to each other or equal to their supplements:
\[
x + 12° = 72° - 2x \quad \text{or} \quad x + 12° = 180° - (72° - 2x)
\]
Solving the first equation:
\[
x + 2x = 72° - 12°
\]
\[
3x = 60°
\]
\[
x = 20°
\]
We will check the second equation:
\[
x + 12° = 108° + 2x
\]
\[
-x = 108° - 12°
\]
\[
-x = 96°
\]
\[
x = -96° \quad (\text{not valid as angles cannot be negative})
\]
Thus, the only solution is:
**x = 20°**
2. The smaller acute angle is \( x = 20° \), and the other angle can be found as follows:
Since the angles in a triangle must sum to \( 90° \) for a right triangle, the other acute angle \( y \) is:
\[
y = 90° - x = 90° - 20° = 70°
\]
Among these, \( 20° \) is the smallest angle.
3. I found the value of \( x \) by setting the two sine expressions equal to each other, solved for \( x \), and then determined the smaller angle in the triangle by subtracting \( x \) from \( 90° \).