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)Responses22°22°36°36°16°16°18°18°

Answer 1

To solve the equation \( \sin(3x + 10^\circ) = \cos(2x - 10^\circ) \), we can use the relationship that states that \( \sin(\theta) = \cos(90^\circ - \theta) \).

Using this property, we can rewrite the equation as:

\[ \sin(3x + 10^\circ) = \sin(90^\circ - (2x - 10^\circ)) \]

Rearranging the right side:

\[ \sin(3x + 10^\circ) = \sin(90^\circ - 2x + 10^\circ) = \sin(100^\circ - 2x) \]

Now, we have:

\[ 3x + 10^\circ = 100^\circ - 2x \]

Next, we solve for \( x \):

1. Add \( 2x \) to both sides: \[ 3x + 2x + 10^\circ = 100^\circ \] \[ 5x + 10^\circ = 100^\circ \]

2. Subtract \( 10^\circ \) from both sides: \[ 5x = 90^\circ \]

3. Divide by 5: \[ x = 18^\circ \]

Thus, the correct value of \( x \) is \( 18^\circ \).

So the answer is \( \boxed{18^\circ} \).