Question

Given the expression sin(3x+10°)=cos(2x−10°)
, which of the following uses the relationship between the sine and cosine of complementary angles to correctly determine the value of x?(1 point)
Responses

36°
36°

16°
16°

22°
22°

18°

Answer 1

To solve the equation \(\sin(3x + 10°) = \cos(2x - 10°)\), we can use the relationship between sine and cosine of complementary angles, which states that \(\sin(A) = \cos(90° - A)\).

Using this relationship, we can re-write the cosine term:

\[
\cos(2x - 10°) = \sin(90° - (2x - 10°)) = \sin(100° - 2x)
\]

So, we can rewrite the original equation as:

\[
\sin(3x + 10°) = \sin(100° - 2x)
\]

Now, if \(\sin(A) = \sin(B)\), then either:

1. \(A = B + k \cdot 180°\) for some integer \(k\), or
2. \(A = 180° - B + k \cdot 180°\) for some integer \(k\).

**Case 1:**
\[
3x + 10° = 100° - 2x + k \cdot 180°
\]
Solving for \(x\):
\[
3x + 2x = 100° - 10° + k \cdot 180°
\]
\[
5x = 90° + k \cdot 180°
\]
\[
x = 18° + k \cdot 36°
\]

**Case 2:**
\[
3x + 10° = 180° - (100° - 2x) + k \cdot 180°
\]
This simplifies to:
\[
3x + 10° = 80° + 2x + k \cdot 180°
\]
Solving for \(x\):
\[
3x - 2x = 80° - 10° + k \cdot 180°
\]
\[
x = 70° + k \cdot 180°
\]

Next, we consider the values of \(x\) obtained from \( k = 0 \):

From the first case, when \(k = 0\):
\[
x = 18°
\]

From the second case, when \(k = 0\):
\[
x = 70°
\]

Given the options \(36°, 16°, 22°, 18°\), the only possible value from our calculations is:

**18°.**