Question

The Sine and Cosine Ratios Quick Check 4 of 54 of 5 Items 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 22° 22° 36° 36° 18° 18° 16° 16° Skip to navigation page 21 of 21

Answer 1

To solve the equation \( \sin(3x + 10^\circ) = \cos(2x - 10^\circ) \), we can use the complementary angle identity that states:

\[ \sin(\theta) = \cos(90^\circ - \theta) \]

In this case, we can express the cosine function in terms of sine:

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

Now, we can set our expressions equal to each other:

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

Since the sine function is equal, we can equate the angles:

1. \( 3x + 10^\circ = 100^\circ - 2x \)

Now, let's solve for \( x \):

\[ 3x + 2x = 100^\circ - 10^\circ \] \[ 5x = 90^\circ \] \[ x = \frac{90^\circ}{5} = 18^\circ \]

Now, let's check if any of the provided responses match \( x = 18^\circ \):

• 22°
• 36°
• 18° ✅
• 16°

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