To find the value of [tex]\(\theta\)[/tex] such that [tex]\(\cos \theta - \cos (90^\circ - \theta) = 0\)[/tex] where [tex]\(\theta\)[/tex] is an acute angle, follow these steps:
1. Recall the Co-function Identity:
We know from trigonometric identities that [tex]\(\cos (90^\circ - \theta) = \sin \theta\)[/tex]. So, we can rewrite the given equation using this identity.
2. Rewrite the Equation:
Substitute the identity into the original equation:
[tex]\[
\cos \theta - \sin \theta = 0
\][/tex]
3. Isolate the Trigonometric Functions:
The equation can be rearranged to isolate the trigonometric functions:
[tex]\[
\cos \theta = \sin \theta
\][/tex]
4. Divide by [tex]\(\cos \theta\)[/tex] (Given that [tex]\(\cos\theta \neq 0\)[/tex]):
We can divide both sides by [tex]\(\cos \theta\)[/tex] to get:
[tex]\[
1 = \tan \theta
\][/tex]
5. Solve for [tex]\(\theta\)[/tex]:
Since [tex]\(\tan \theta = 1\)[/tex], we find the angle [tex]\(\theta\)[/tex] whose tangent is 1. The tangent of 45 degrees is 1. Therefore:
[tex]\[
\theta = 45^\circ
\][/tex]
6. Check for Acute Angle:
An acute angle is between 0 and 90 degrees. The angle [tex]\(\theta = 45^\circ\)[/tex] fits within this range.
So, the value of [tex]\(\theta\)[/tex] that satisfies the equation and is acute is:
[tex]\[
\theta = 45^\circ
\][/tex]
