Solveeit Logo
Login

Question

Question: The value of \(\sin\theta + \cos\theta\) will be greatest when...

The value of sin⁡θ+cos⁡θ\sin\theta + \cos\theta will be greatest when

A

θ=30o\theta = 30^{o}

B

θ=45o\theta = 45^{o}

C

θ=60o\theta = 60^{o}

D

θ=90o\theta = 90^{o}

Answer

θ=45o\theta = 45^{o}

Explanation

Solution

Let f(x)=sin⁡θ+cos⁡θ=2sin⁡(θ+π4)−1≤sin⁡(θ+π4)≤1f(x) = \sin\theta + \cos\theta = \sqrt{2}\sin(\theta + \frac{\pi}{4}) - 1 \leq \sin(\theta + \frac{\pi}{4}) \leq 1

⇒ −2≤2sin⁡(θ+π4)≤2- \sqrt{2} \leq \sqrt{2}\sin(\theta + \frac{\pi}{4}) \leq \sqrt{2}

If f(x)f(x) is maximum then,

sin⁡(θ+π4)=1=sin⁡π2\sin(\theta + \frac{\pi}{4}) = 1 = \sin\frac{\pi}{2} ⇒ θ=π4\theta = \frac{\pi}{4} ⇒ θ+π4=π2⇒θ=π4\theta + \frac{\pi}{4} = \frac{\pi}{2} \Rightarrow \theta = \frac{\pi}{4}.