To solve the equation \(\cos (50 + x)^\circ = \sin (2x - 6)^\circ\), we can use trigonometric identities to simplify and solve for \( x \).
1. Use the identity: \(\sin \theta = \cos (90^\circ - \theta)\)
Therefore, \(\sin (2x - 6)^\circ = \cos \left(90^\circ - (2x - 6)^\circ\right) = \cos (96^\circ - 2x)^\circ\).
2. Rewrite the equation:
[tex]\[
\cos (50 + x)^\circ = \cos (96 - 2x)^\circ
\][/tex]
3. Equate the angles:
For two cosines to be equal, the angles must be congruent to each other modulo \(360^\circ\), so we have:
[tex]\[
50 + x = 96 - 2x + 360k \quad \text{or} \quad 50 + x = -96 + 2x + 360k
\][/tex]
for \(k \in \mathbb{Z}\).
4. Solve the first equation:
[tex]\[
50 + x = 96 - 2x + 360k
\][/tex]
[tex]\[
3x = 46 + 360k
\][/tex]
[tex]\[
x = \frac{46 + 360k}{3}
\][/tex]
For \(k = 0\):
[tex]\[
x = \frac{46}{3} \approx 15.3
\][/tex]
For \(k = 1\):
[tex]\[
x = \frac{406}{3} \approx 135.3
\][/tex]
For \(k = -1\):
[tex]\[
x = \frac{-314}{3} \approx -104.7
\][/tex]
5. Solve the second equation:
[tex]\[
50 + x = -96 + 2x + 360k
\][/tex]
[tex]\[
50 + x = -96 + 2x + 360k
\][/tex]
[tex]\[
50 + 96 = x + 360k
\][/tex]
[tex]\[
x = 146 + 360k
\][/tex]
For \(k = 0\):
[tex]\[
x = 146
\][/tex]
6. Check the values for each \(k\), and realistic solutions:
We are only interested in the values \(0 \leq x < 360\).
Therefore, the candidates \( x \approx 15.3, 135.3, -104.7, 146 \). Within \(0 \leq x < 360\), appropriate values are \( 15.3 \) and \(146\).
7. Compare with given choices \(0, 46, 15.3, 44, 18.7^\circ \):
The nearest tenth from our solution set is \(15.3 \).
Therefore, the value of [tex]\( x \)[/tex] to the nearest tenth is [tex]\(15.3^\circ\)[/tex].
