Question

If α+β =90° and If If α+β =90° and α : β = 2:1 find the value of sinα : sinβ​

Answer 1

Question :

If If α+β =90° and α : β = 2:1 find the value of sinα : sinβ.

Solution :

We have ,α+β =90°

⇒β =90°-α....(1)

and ,α : β = 2:1

$\sf\implies\dfrac{\alpha}{90-\alpha}=\dfrac{2}{1}$

$\sf\implies\alpha=180-2\alpha$

$\sf\implies3\alpha=180$

$\sf\implies\alpha=60$

Thus,β =90°-α

⇒β =90°-60°=30°

Therefore ,

$\sf\implies\dfrac{\sin\alpha}{\sin\beta}$

$\sf=\dfrac{\sin60\degree}{\sin30\degree}$

$\sf=\dfrac{\sqrt{3}}{1}$

Therefore,sinα : sinβ= √3:1

Answer 2

Given -

[Let alpha = a , Beta = b ]

→ a + b = 90°

→ a : b = 2 : 1

To Find :-

→ Value of Sin a : Sin B

Solution :-

→ a + b = 90

Let a = 2x and b = X .

Sum of a + b will be :-

→ 2x + X = 3x.

$\sf{\implies a = \frac{2x}{3x} \times 90 }$

$\sf{\implies a = 2 \times 30 = 60 }$

→ a + b = 90

→ 60 + b = 90

→ b = 30 °

Putting value in Sin a and sin B -

$\implies \: \frac{ \sin(a) }{ \sin(b) } = \frac{ \sin(60) }{ \sin(30) }$

$\implies \: \sin 60= \frac{ \sqrt{3} }{ 2 }$

$\implies \: \sin 30= \frac{ 1 }{ 2 }$

Putting these values in ratio -

$\implies \: \frac{ \sin(a) }{ \sin(b) } = \frac{ \frac{ \sqrt{3} }{2} }{ \frac{1}{2} }$

$\implies \: \frac{ \sin(a) }{ \sin(b) } = \frac{ \sqrt{3} }{2} \times 2$

Sin a : Sin B = √3 : 1