Question

If sin theta -cos theta =1/2.then find the value of 1/sintheta+cos theta

Answer 1

Answer:

$\frac{1}{sinA+cosA}=\frac{2}{\sqrt{7} }$

Step-by-step explanation:

First we need to know the following TRIGONOMETRIC IDENTITY:

$sin^{2}A+cos^{2}A=1$

Now we need to find an equality on our first equation.

If, $sinA-cosA=\frac{1}{2}$, we can square both sides of the equation resulting in:

$sin^{2}A-2sinAcosA+cos^{2}A=\frac{1}{4}   \\ sin^{2}A+cos^{2}A-2sinAcosA=\frac{1}{4}$

Replacing the TRIGONOMETRIC IDENTITY:

$1-2sinAcosA=\frac{1}{4} \\ -2sinAcosA=\frac{1}{4}-1\\ 2sinAcosA=1-\frac{1}{4}$

------------------------------------------------------------------------------------------------

Now we need to transform the second equation, REMEBER we can square a number and root-extract it at the same time without altering its value.

$\sqrt{\frac{1}{(sinA+cosA)^{2} } } =\sqrt{\frac{1}{(sin^{2}A+2sinAcosA+cos^{2}A)  } }=\sqrt{\frac{1}{(sin^{2}A+cos^{2}A+2sinAcosA) } }$

Now we can use the equality we found previously and the TRIGONOMETRIC IDENTITY to find our answer:

$\sqrt{\frac{1}{(sin^{2}A+cos^{2}A+2sinAcosA) } }=\sqrt{\frac{1}{1+\frac{3}{4} } }= \sqrt{\frac{1}{(\frac{7}{4}) } }=\sqrt{\frac{4}{7} }=\frac{2}{\sqrt{7} }$