Solve 10/sin(75^circ)=c/sqrt{2/2}

$$\frac{ 10 }{ \sin ( 75 ^ { \circ } ) } = \frac{ c }{ \frac{ \sqrt{ 2 } }{ 2 } }$$

Solve for c

$$c = \frac{50000000000000000 \sqrt{2}}{9659258262890683} = 7\frac{3095870278419972}{9659258262890684} \approx 7.320508076$$

Steps for Solving Linear Equation

Evaluate trigonometric functions in the problem

$$\frac{10}{0.9659258262890683} = \frac{c}{\frac{\sqrt{2}}{2}}$$

Expand \frac{10}{0.9659258262890683} by multiplying both numerator and the denominator by 10000000000000000.

$$\frac{100000000000000000}{9659258262890683}=\frac{c}{\frac{\sqrt{2}}{2}}$$

Divide c by \frac{\sqrt{2}}{2} by multiplying c by the reciprocal of \frac{\sqrt{2}}{2}.

$$\frac{100000000000000000}{9659258262890683}=\frac{c\times 2}{\sqrt{2}}$$

Rationalize the denominator of \frac{c\times 2}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{100000000000000000}{9659258262890683}=\frac{c\times 2\sqrt{2}}{\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{100000000000000000}{9659258262890683}=\frac{c\times 2\sqrt{2}}{2}$$

Cancel out 2 and 2.

$$\frac{100000000000000000}{9659258262890683}=c\sqrt{2}$$

Swap sides so that all variable terms are on the left hand side.

$$c\sqrt{2}=\frac{100000000000000000}{9659258262890683}$$

The equation is in standard form.

$$\sqrt{2}c=\frac{100000000000000000}{9659258262890683}$$

Divide both sides by \sqrt{2}.

$$\frac{\sqrt{2}c}{\sqrt{2}}=\frac{\frac{100000000000000000}{9659258262890683}}{\sqrt{2}}$$

Dividing by \sqrt{2} undoes the multiplication by \sqrt{2}.

$$c=\frac{\frac{100000000000000000}{9659258262890683}}{\sqrt{2}}$$

Divide \frac{100000000000000000}{9659258262890683} by \sqrt{2}.

$$c=\frac{50000000000000000\sqrt{2}}{9659258262890683}$$