To solve the equation [tex]\(\cos x = \sin (20 + x)^\circ\)[/tex] with the condition [tex]\(0^\circ < x < 90^\circ\)[/tex], follow these steps:
1. Understanding Trigonometric Identities:
Recall that [tex]\(\sin (90^\circ - \theta) = \cos \theta\)[/tex]. This identity helps simplify trigonometric equations by converting sine functions to cosine functions or vice versa.
2. Set Up the Equation:
Given [tex]\(\cos x = \sin (20 + x)^\circ\)[/tex], use the identity [tex]\(\sin (90^\circ - \theta)\)[/tex] to rewrite [tex]\(\sin (20 + x)^\circ\)[/tex]:
[tex]\[
\cos x = \sin (20 + x)^\circ
\][/tex]
[tex]\[
\cos x = \cos (90 - (20 + x))^\circ
\][/tex]
Simplify the argument of the cosine on the right-hand side:
[tex]\[
\cos x = \cos (70 - x)^\circ
\][/tex]
3. Equating the Angles:
Since [tex]\(\cos \theta = \cos \phi\)[/tex] implies that [tex]\(\theta = 360^\circ n \pm \phi\)[/tex] for [tex]\(n \in \mathbb{Z}\)[/tex] and given that [tex]\(0^\circ < x < 90^\circ\)[/tex], we focus on:
[tex]\[
x = 70 - x
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Add [tex]\(x\)[/tex] to both sides of the equation to isolate [tex]\(x\)[/tex] on one side:
[tex]\[
x + x = 70^\circ
\][/tex]
[tex]\[
2x = 70^\circ
\][/tex]
[tex]\[
x = 35^\circ
\][/tex]
5. Check the Interval:
Verify that the solution [tex]\(x = 35^\circ\)[/tex] lies within the given range [tex]\(0^\circ < x < 90^\circ\)[/tex]. Since [tex]\(35^\circ\)[/tex] falls within this interval, it is a valid solution.
Thus, the value of [tex]\(x\)[/tex] is [tex]\(35\)[/tex].
