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° 22° 22° 16° 16° 18° 18°

Answer 1

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

Thus, we can rewrite the equation as:

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

This simplifies to:

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

Now, we can set the angles equal to each other or their supplements (since \( \sin(A) = \sin(B) \) implies \( A = B + n \cdot 180° \) or \( A = 180° - B + n \cdot 180° \), where \( n \) is an integer).

1. Setting the angles equal:

\[ 3x + 10° = 100° - 2x \]

Now, solve for \( x \):

\[ 3x + 2x = 100° - 10° \]

\[ 5x = 90° \]

\[ x = \frac{90°}{5} = 18° \]

2. Now let’s check for the supplements:

\[ 3x + 10° = 180° - (100° - 2x) \]

Simplifying, we have:

\[ 3x + 10° = 180° - 100° + 2x \]

\[ 3x + 10° = 80° + 2x \]

Subtracting \( 2x \) from both sides:

\[ x + 10° = 80° \]

\[ x = 80° - 10° = 70° \]

Since \( x = 70° \) does not fall within the options provided, we will take \( x = 18° \) as it is the only valid solution we found.

Therefore, the correct answer is 18°.