Answer :
Let's solve the equation [tex]\(\cos x = \sin (20^\circ + x)\)[/tex] given the constraint [tex]\(0^\circ < x < 90^\circ\)[/tex].
1. Recall the trigonometric identity:
[tex]\[ \sin(\theta) = \cos(90^\circ - \theta) \][/tex]
2. Applying this identity to the given equation:
[tex]\[ \cos x = \sin (20^\circ + x) \][/tex]
can be rewritten using the identity as:
[tex]\[ \cos x = \cos(90^\circ - (20^\circ + x)) \][/tex]
3. Simplify the expression inside the cosine:
[tex]\[ \cos x = \cos(90^\circ - 20^\circ - x) \][/tex]
[tex]\[ \cos x = \cos(70^\circ - x) \][/tex]
4. Since cosine is an even function and [tex]\( \cos A = \cos B \)[/tex] implies that [tex]\( A = B \)[/tex], we have:
[tex]\[ x = 70^\circ - x \][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + x = 70^\circ \][/tex]
[tex]\[ 2x = 70^\circ \][/tex]
[tex]\[ x = \frac{70^\circ}{2} \][/tex]
[tex]\[ x = 35^\circ \][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\( \boxed{35} \)[/tex].
1. Recall the trigonometric identity:
[tex]\[ \sin(\theta) = \cos(90^\circ - \theta) \][/tex]
2. Applying this identity to the given equation:
[tex]\[ \cos x = \sin (20^\circ + x) \][/tex]
can be rewritten using the identity as:
[tex]\[ \cos x = \cos(90^\circ - (20^\circ + x)) \][/tex]
3. Simplify the expression inside the cosine:
[tex]\[ \cos x = \cos(90^\circ - 20^\circ - x) \][/tex]
[tex]\[ \cos x = \cos(70^\circ - x) \][/tex]
4. Since cosine is an even function and [tex]\( \cos A = \cos B \)[/tex] implies that [tex]\( A = B \)[/tex], we have:
[tex]\[ x = 70^\circ - x \][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[ x + x = 70^\circ \][/tex]
[tex]\[ 2x = 70^\circ \][/tex]
[tex]\[ x = \frac{70^\circ}{2} \][/tex]
[tex]\[ x = 35^\circ \][/tex]
Therefore, the value of [tex]\(x\)[/tex] is [tex]\( \boxed{35} \)[/tex].
