Solve sin10^circ/1-sqrt{3tan10^circ}

$$\frac { \sin 10 ^ { \circ } } { 1 - \sqrt { 3 } \tan 10 ^ { \circ } }$$

Evaluate

$$\frac{153094294367684618385708214524217\sqrt{3}}{4533631938113549281696348581703994}+\frac{434120444167325825000000000000000}{2266815969056774640848174290851997}\approx 0.25$$

Solution Steps

Evaluate trigonometric functions in the problem

$$\frac{0.17364817766693033}{1 - \sqrt{3} 0.17632698070846498}$$

Rationalize the denominator of \frac{0.17364817766693033}{1-0.17632698070846498\sqrt{3}} by multiplying numerator and denominator by 1+0.17632698070846498\sqrt{3}.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{\left(1-0.17632698070846498\sqrt{3}\right)\left(1+0.17632698070846498\sqrt{3}\right)}$$

Consider \left(1-0.17632698070846498\sqrt{3}\right)\left(1+0.17632698070846498\sqrt{3}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1^{2}-\left(-0.17632698070846498\sqrt{3}\right)^{2}}$$

Calculate 1 to the power of 2 and get 1.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1-\left(-0.17632698070846498\sqrt{3}\right)^{2}}$$

Expand \left(-0.17632698070846498\sqrt{3}\right)^{2}.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1-\left(-0.17632698070846498\right)^{2}\left(\sqrt{3}\right)^{2}}$$

Calculate -0.17632698070846498 to the power of 2 and get 0.0310912041257633812202434278864004.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1-0.0310912041257633812202434278864004\left(\sqrt{3}\right)^{2}}$$

The square of \sqrt{3} is 3.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1-0.0310912041257633812202434278864004\times 3}$$

Multiply 0.0310912041257633812202434278864004 and 3 to get 0.0932736123772901436607302836592012.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{1-0.0932736123772901436607302836592012}$$

Subtract 0.0932736123772901436607302836592012 from 1 to get 0.9067263876227098563392697163407988.

$$\frac{0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right)}{0.9067263876227098563392697163407988}$$

Divide 0.17364817766693033\left(1+0.17632698070846498\sqrt{3}\right) by 0.9067263876227098563392697163407988 to get \frac{434120444167325825000000000000000}{2266815969056774640848174290851997}\left(1+0.17632698070846498\sqrt{3}\right).

$$\frac{434120444167325825000000000000000}{2266815969056774640848174290851997}\left(1+0.17632698070846498\sqrt{3}\right)$$

Use the distributive property to multiply \frac{434120444167325825000000000000000}{2266815969056774640848174290851997} by 1+0.17632698070846498\sqrt{3}.

$$\frac{434120444167325825000000000000000}{2266815969056774640848174290851997}+\frac{153094294367684618385708214524217}{4533631938113549281696348581703994}\sqrt{3}$$