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Question

Question: 3cos36+5sin18/5cos36-3sin18...

3cos36+5sin18/5cos36-3sin18

Answer

175−2411\frac{17\sqrt{5} - 24}{11}

Explanation

Solution

We use the standard values for sin⁡(18∘)=5−14\sin(18^\circ) = \frac{\sqrt{5}-1}{4} and cos⁡(36∘)=5+14\cos(36^\circ) = \frac{\sqrt{5}+1}{4}. Substituting these values into the expression: Numerator: 3(5+14)+5(5−14)=35+3+55−54=85−24=45−123\left(\frac{\sqrt{5}+1}{4}\right) + 5\left(\frac{\sqrt{5}-1}{4}\right) = \frac{3\sqrt{5}+3 + 5\sqrt{5}-5}{4} = \frac{8\sqrt{5}-2}{4} = \frac{4\sqrt{5}-1}{2}. Denominator: 5(5+14)−3(5−14)=55+5−35+34=25+84=5+425\left(\frac{\sqrt{5}+1}{4}\right) - 3\left(\frac{\sqrt{5}-1}{4}\right) = \frac{5\sqrt{5}+5 - 3\sqrt{5}+3}{4} = \frac{2\sqrt{5}+8}{4} = \frac{\sqrt{5}+4}{2}. The expression becomes 45−125+42=45−15+4\frac{\frac{4\sqrt{5}-1}{2}}{\frac{\sqrt{5}+4}{2}} = \frac{4\sqrt{5}-1}{\sqrt{5}+4}. Rationalizing the denominator: 45−15+4×5−45−4=20−165−5+45−16=24−175−11=175−2411\frac{4\sqrt{5}-1}{\sqrt{5}+4} \times \frac{\sqrt{5}-4}{\sqrt{5}-4} = \frac{20 - 16\sqrt{5} - \sqrt{5} + 4}{5 - 16} = \frac{24 - 17\sqrt{5}}{-11} = \frac{17\sqrt{5} - 24}{11}.