Question

If sin alpha = 1/root 10 and since beta = 1/ root 5.alpha and beta are acute angles then show that alpha + beta = pi/4

Answer 1

Step-by-step explanation:

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Answer 2

Concept Introduction:

The study of trigonometry deals with the relationship between side lengths and angles in triangles.

Working formula is:- sin(α+β) = sinαcosβ + cosαsinβ

Given:

Value of sinα = $\frac{1}{\sqrt{10} }$  and sinβ = $\frac{1}{\sqrt{5} }$

To Find:

We have to show that α+β = $\frac{\pi }{4}$

Solution:

According to the problem,

sinα=$\frac{1}{\sqrt{10}}$ , ∴ cosα = $\sqrt[2]{1-sin^{2} \alpha }$ = $\sqrt{1-\frac{1}{10} }$ = $\sqrt{\frac{9}{10} }$ =

sinβ=$\frac{1}{\sqrt{5}}$ ∴ cosβ  = $\sqrt[2]{1-sin^{2} \beta }$ = $\sqrt{1-\frac{1}{5} }$ = $\sqrt{\frac{4}{5} }$ = $\frac{2}{\sqrt{5}}$

sin(α+β) = sinαcosβ + cosαsinβ = $\frac{1}{\sqrt{10}}$ × $\frac{2}{\sqrt{5}}$ + $\frac{3}{\sqrt{10}}$ ×$\frac{1}{\sqrt{5}}$ = $\frac{5}{5\sqrt{2}}$ = $\frac{1}{\sqrt{2}}$

⇒$\frac{1}{\sqrt{2}}$ = sin($\frac{\pi }{4}$)

⇒ α+β = $\frac{\pi }{4}$

Hence proved.

Final Answer:

The value of α+β is  $\frac{\pi }{4}$ .

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