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Trigonometric Identities

If sinθ+cosθ=...

Question

If $sin θ + cos θ = 1$ , then find the general value of $θ$ .

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Solution

Given:

$sin θ + cos θ = 1$
Dividing both sides by $\sqrt{1^{2} + 1^{2}} = \sqrt{2}$ ,
we get
$\Rightarrow \frac{1}{\sqrt{2}} sin θ + \frac{1}{\sqrt{2}} cos θ = \frac{1}{\sqrt{2}}$
$\Rightarrow cos (\frac{π}{4}) sin θ + sin (\frac{π}{4}) cos θ = sin (\frac{π}{4})$
$\Rightarrow sin (θ + \frac{π}{4}) = sin \frac{π}{4}$
$[_{= sin (A + B)}^{∵ sin A cos B + cos A sin B}]$
$\Rightarrow θ + \frac{π}{4} = n π + (- 1)^{n} \frac{π}{4}, n ϵ Z$
$[_{θ = n π + (- 1)^{n} α, n ϵ Z}^{If sin θ = sin α, then}]$
$\Rightarrow θ = n π + (- 1)^{n} \frac{π}{4} - \frac{π}{4}, n ϵ Z$
Hence, general value is
$θ = n π + (- 1)^{n} \frac{π}{4} - \frac{π}{4}, n ϵ Z$

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