Question

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Subject : Maths
Chapter : Trigonometry

The equation sin² x = (a²+b²)/(2ab), is possible if

(a) a = b
(b) a = -b
(c) 2a = b
(d) None of these ​

Answer 1

Answer:

a=b

Step-by-step explanation:

Given equation is sin²x=(a²+b²)/(2ab) is possible for a=b

1)If a=b then

sin²x=(b²+b²)/(2×b×b)

=>sin²x=2b²/2b²

=>sin²x=1

=>(sinx)²=1

=>sin x=1

=>Sin x=Sin 90°

x=90°

2) if a=-b then

sin² x=((-b)²+b²)/(2×-b×b)

=>sin² x=2b²/-2b²

=>sin² x=-1

=>sin x=√-1

so the square root of -1 is not a real number.

3) If 2a =b then

sin² x=(a²+(2a)²)/(2×a×2a)

=>sin² x=(a²+4a²)/(4a²)

=>sin² x=5a²/4a²

=>sin² x=5/4

=>sin x=√5/4

so,The equation sin² x = (a²+b²)/(2ab), is possible if a=b

Answer 2

Given :

$\displaystyle {\sf sin^2 x=\frac{(a^2+b^2)}{2ab}}$ is possible if :

• $\sf{a=b}$
• $\sf{a=-b}$
• $\sf{2a=b}$
• $\sf{None\:of\:these}$

Solution :

$\large{\sf{a)a=b}}$

• Substitute the value of a or b in place of a or b in the given equation.

$\implies \displaystyle {\sf sin^2 x= \frac{(a^2+b^2)}{2ab}}$

$\implies \displaystyle \sf{sin ^2x = \frac{a^2+a^2 }{2 \times a \times a}}$

$\implies \displaystyle {\sf sin^2 x = \frac{\cancel{2a^2 }}{\cancel{2a^2}}}$

$\implies \sf sin^2 x = 1$

$\implies \sf sin^2 x= 1$

$\boxed{\sf sinx = 90 ^{\circ}}$

____________________________________

$\large{\sf{ b)a=-b}}$

• Substitute -b in place of a in the given equation.

$\implies \displaystyle {\sf sin^2 x=\frac{(a^2+b^2)}{2ab}}$

$\implies \displaystyle {\sf sin^2 x=\frac{\Big((-b)^2+(b)^2\Big)}{2\times -b \times b}}$

$\implies \displaystyle {\sf sin^2 x = \frac{\cancel{2b^2 }}{- \cancel{2b^2}}}$

$\implies \sf sin^2 x = -1$

$\boxed{ \sf{ sin x = \sqrt{-1}}}$

____________________________________

$\large{\sf{c) 2a=b}}$

• Substitute 2a in place of b in the given equation.

$\implies \displaystyle {\sf sin^2 x=\frac{(a^2+b^2)}{2ab}}$

$\implies \displaystyle {\sf sin^2 x=\frac{\Big(a^2+(2a)^2\Big)}{2 \times a \times 2a}}$

$\implies \displaystyle {\sf sin^2 x=\frac{(a^2+4a^2)}{4a^2 }}$

$\implies \displaystyle {\sf sin^2 x=\frac{(5\cancel{a^2} )}{4\cancel {a^2}}}$

$\implies \displaystyle {\sf sin^2 x = \frac{5}{4}}$

$\displaystyle {\boxed{\sf sin x = \frac{\sqrt{5}}{2} }}$

____________________________________

Required Answer :

The equation $\displaystyle {\sf sin^2 x=\frac{(a^2+b^2)}{2ab}}$ is possible if $\sf{\underline {a=b}}$