Answer :
Given the equation:
[tex]\[ \cos (50 + x) = \sin (2x - 6) \][/tex]
We can use the identity for cosine and sine:
[tex]\[ \cos \theta = \sin (90^\circ - \theta) \][/tex]
Applying this identity to \(\cos (50 + x)\):
[tex]\[ \cos (50 + x) = \sin (90^\circ - (50 + x)) = \sin (40 - x) \][/tex]
Thus the given equation becomes:
[tex]\[ \sin (40 - x) = \sin (2x - 6) \][/tex]
For two sine functions to be equal:
[tex]\[ \sin A = \sin B \implies A = B + 360^\circ k \quad \text{or} \quad A = 180^\circ - B + 360^\circ k \quad \text{for some integer } k \][/tex]
So, we have two cases to consider:
### Case 1: Same angle
[tex]\[ 40 - x = 2x - 6 \][/tex]
Solving for \(x\):
[tex]\[ 40 + 6 = 2x + x \][/tex]
[tex]\[ 46 = 3x \][/tex]
[tex]\[ x = \frac{46}{3} \][/tex]
[tex]\[ x \approx 15.3 \][/tex]
### Case 2: Supplementary angle
[tex]\[ 40 - x = 180^\circ - (2x - 6) \][/tex]
[tex]\[ 40 - x = 180 - 2x + 6 \][/tex]
[tex]\[ 40 - x = 186 - 2x \][/tex]
Rearrange the equation:
[tex]\[ 40 - 186 = -2x + x \][/tex]
[tex]\[ -146 = -x \][/tex]
[tex]\[ x = 146 \][/tex]
Now, examining the range, \(x = 146\) degrees doesn't align with typical angle measures used in trigonometric solutions within the context of typical questions.
Thus, reevaluating the practical and possible answers:
The only possible valid solution (considering usual angle restrictions in trigonometric equations) is:
[tex]\[ x \approx 15.3 \][/tex]
Among the given options:
46, 18.7, 44, 15.3
The correct answer is:
[tex]\[ \boxed{15.3} \][/tex]
[tex]\[ \cos (50 + x) = \sin (2x - 6) \][/tex]
We can use the identity for cosine and sine:
[tex]\[ \cos \theta = \sin (90^\circ - \theta) \][/tex]
Applying this identity to \(\cos (50 + x)\):
[tex]\[ \cos (50 + x) = \sin (90^\circ - (50 + x)) = \sin (40 - x) \][/tex]
Thus the given equation becomes:
[tex]\[ \sin (40 - x) = \sin (2x - 6) \][/tex]
For two sine functions to be equal:
[tex]\[ \sin A = \sin B \implies A = B + 360^\circ k \quad \text{or} \quad A = 180^\circ - B + 360^\circ k \quad \text{for some integer } k \][/tex]
So, we have two cases to consider:
### Case 1: Same angle
[tex]\[ 40 - x = 2x - 6 \][/tex]
Solving for \(x\):
[tex]\[ 40 + 6 = 2x + x \][/tex]
[tex]\[ 46 = 3x \][/tex]
[tex]\[ x = \frac{46}{3} \][/tex]
[tex]\[ x \approx 15.3 \][/tex]
### Case 2: Supplementary angle
[tex]\[ 40 - x = 180^\circ - (2x - 6) \][/tex]
[tex]\[ 40 - x = 180 - 2x + 6 \][/tex]
[tex]\[ 40 - x = 186 - 2x \][/tex]
Rearrange the equation:
[tex]\[ 40 - 186 = -2x + x \][/tex]
[tex]\[ -146 = -x \][/tex]
[tex]\[ x = 146 \][/tex]
Now, examining the range, \(x = 146\) degrees doesn't align with typical angle measures used in trigonometric solutions within the context of typical questions.
Thus, reevaluating the practical and possible answers:
The only possible valid solution (considering usual angle restrictions in trigonometric equations) is:
[tex]\[ x \approx 15.3 \][/tex]
Among the given options:
46, 18.7, 44, 15.3
The correct answer is:
[tex]\[ \boxed{15.3} \][/tex]
