Find the value of \(\theta\) if \(\sin2\theta=\cos45^\circ\sin45^\circ+\sin30^\circ\)
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(45^\circ\)
The right-hand side uses only standard angle values, so it is worked out first; \(\theta\) then comes from the angle whose sine equals that value.
Both 45° values are \(\frac{1}{\sqrt{2}}\), so their product is \(\cos45^\circ\sin45^\circ=\frac{1}{\sqrt{2}}\times\frac{1}{\sqrt{2}}=\frac{1}{2}\)
The standard value \(\sin30^\circ=\frac{1}{2}\)
Right-hand side: \(\sin2\theta=\frac{1}{2}+\frac{1}{2}=1\)
The sine is 1 at 90°, so \(2\theta=90^\circ\) and \(\theta=45^\circ\)