Exercise 34 1. If \( \cos \theta=\sin \theta \). Find the value of \( \theta \) Solution

To solve the equation \( \cos \theta = \sin \theta \), we can start by dividing both sides by \( \cos \theta \) (assuming \( \cos \theta \neq 0 \)), which gives us \( 1 = \tan \theta \). The tangent function equals 1 at specific angles, namely \( \theta = 45^\circ + k \cdot 180^\circ \), where \( k \) is any integer. Therefore, \( \theta \) can be expressed as \( \theta = 45^\circ + k \cdot 180^\circ \). Alternatively, we can also use the unit circle to find the angles where sine and cosine are equal. The angle \( \theta = 45^\circ \) or \( \pi/4 \) radians is the first quadrant solution, and due to the periodic nature of trigonometric functions, there are infinitely many solutions in the format of \( \theta = 45^\circ + k \cdot 180^\circ \). This should yield the result nicely!