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)

Question

Responses

36°
36°

22°
22°

18°
18°

16°

Answer 1

To solve the equation \( \sin(3x + 10°) = \cos(2x - 10°) \) using the relationship between sine and cosine for complementary angles, we can use the fact that \( \sin(θ) = \cos(90° - θ) \).

So, \[ \sin(3x + 10°) = \cos(2x - 10°) \implies 3x + 10° = 90° - (2x - 10°) \]

Now, simplify the equation: \[ 3x + 10° = 90° - 2x + 10° \] \[ 3x + 10° = 100° - 2x \]

Now, add \( 2x \) to both sides: \[ 3x + 2x + 10° = 100° \] \[ 5x + 10° = 100° \]

Next, subtract \( 10° \) from both sides: \[ 5x = 90° \]

Finally, divide by 5: \[ x = 18° \]

Thus, the correct answer is: 18°.