Question
For the acute angles in a right triangle, sin(3x+10°)=cos(x+24°)
sin
(
3
𝑥
+
10
°
)
=
cos
(
𝑥
+
24
°
)
. What is the value of x
𝑥
?(1 point)
Responses
31°
31
°
31 degrees
14°
14
°
14 degrees
28°
28
°
28 degrees
16°
sin
(
3
𝑥
+
10
°
)
=
cos
(
𝑥
+
24
°
)
. What is the value of x
𝑥
?(1 point)
Responses
31°
31
°
31 degrees
14°
14
°
14 degrees
28°
28
°
28 degrees
16°
Answers
GPT-4o mini
To solve the equation \( \sin(3x + 10^\circ) = \cos(x + 24^\circ) \), we can use the complementary angle identity, which states that \( \sin(A) = \cos(90^\circ - A) \). This gives us:
\[
\sin(3x + 10^\circ) = \cos(x + 24^\circ) \implies 3x + 10^\circ = 90^\circ - (x + 24^\circ)
\]
Now we can simplify the right side:
\[
3x + 10^\circ = 90^\circ - x - 24^\circ
\]
Combine like terms on the right side:
\[
3x + 10^\circ = 66^\circ - x
\]
Next, we'll move \( x \) to the left side and \( 10^\circ \) to the right side:
\[
3x + x = 66^\circ - 10^\circ
\]
\[
4x = 56^\circ
\]
Now we divide by 4 to solve for \( x \):
\[
x = \frac{56^\circ}{4} = 14^\circ
\]
Thus, the value of \( x \) is \( 14^\circ \).
The answer is \( \boxed{14^\circ} \).
\[
\sin(3x + 10^\circ) = \cos(x + 24^\circ) \implies 3x + 10^\circ = 90^\circ - (x + 24^\circ)
\]
Now we can simplify the right side:
\[
3x + 10^\circ = 90^\circ - x - 24^\circ
\]
Combine like terms on the right side:
\[
3x + 10^\circ = 66^\circ - x
\]
Next, we'll move \( x \) to the left side and \( 10^\circ \) to the right side:
\[
3x + x = 66^\circ - 10^\circ
\]
\[
4x = 56^\circ
\]
Now we divide by 4 to solve for \( x \):
\[
x = \frac{56^\circ}{4} = 14^\circ
\]
Thus, the value of \( x \) is \( 14^\circ \).
The answer is \( \boxed{14^\circ} \).