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Trigonometric Ratio and Identites question

2024 · 8 Apr · Shift 2 · Q48
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Trigonometric Ratio and Identites question

2024 · 8 Apr · Shift 2 · Q48

JEE MainMathematicsTrigonometric Ratio and IdentitesMCQ+4 / −1
If the value of 3cos⁡36∘+5sin⁡18∘5cos⁡36∘−3sin⁡18∘\frac{3 \cos 36^{\circ}+5 \sin 18^{\circ}}{5 \cos 36^{\circ}-3 \sin 18^{\circ}}5cos36∘−3sin18∘3cos36∘+5sin18∘​ is a5−bc\frac{a \sqrt{5}-b}{c}ca5​−b​, where a,b,ca, b, ca,b,c are natural numbers and gcd⁡(a,c)=1\operatorname{gcd}(a, c)=1gcd(a,c)=1, then a+b+ca+b+ca+b+c is equal to :
  1. A
    54
  2. B
    52
  3. C
    50
  4. D
    40
View written solutionFree

Correct answer: B

  1. Use the standard exact values

    We know that cos⁡36∘=5+14,sin⁡18∘=5−14.\cos 36^\circ = \frac{\sqrt{5}+1}{4}, \qquad \sin 18^\circ = \frac{\sqrt{5}-1}{4}.cos36∘=45​+1​,sin18∘=45​−1​.

  2. Substitute into the expression

    Let E=3cos⁡36∘+5sin⁡18∘5cos⁡36∘−3sin⁡18∘.E=\frac{3\cos 36^\circ+5\sin 18^\circ}{5\cos 36^\circ-3\sin 18^\circ}.E=5cos36∘−3sin18∘3cos36∘+5sin18∘​.

    Substituting, E=3⋅5+14+5⋅5−145⋅5+14−3⋅5−14.E=\frac{3\cdot \frac{\sqrt{5}+1}{4}+5\cdot \frac{\sqrt{5}-1}{4}}{5\cdot \frac{\sqrt{5}+1}{4}-3\cdot \frac{\sqrt{5}-1}{4}}.E=5⋅45​+1​−3⋅45​−1​3⋅45​+1​+5⋅45​−1​​.

    Taking 14\frac{1}{4}41​ common from numerator and denominator, E=3(5+1)+5(5−1)5(5+1)−3(5−1).E=\frac{3(\sqrt{5}+1)+5(\sqrt{5}-1)}{5(\sqrt{5}+1)-3(\sqrt{5}-1)}.E=5(5​+1)−3(5​−1)3(5​+1)+5(5​−1)​.

  3. Simplify numerator and denominator

    Numerator: 3(5+1)+5(5−1)=35+3+55−5=85−2.3(\sqrt{5}+1)+5(\sqrt{5}-1)=3\sqrt{5}+3+5\sqrt{5}-5=8\sqrt{5}-2.3(5​+1)+5(5​−1)=35​+3+55​−5=85​−2.

    Denominator: 5(5+1)−3(5−1)=55+5−35+3=25+8.5(\sqrt{5}+1)-3(\sqrt{5}-1)=5\sqrt{5}+5-3\sqrt{5}+3=2\sqrt{5}+8.5(5​+1)−3(5​−1)=55​+5−35​+3=25​+8.

    So, E=85−225+8=45−15+4.E=\frac{8\sqrt{5}-2}{2\sqrt{5}+8} = \frac{4\sqrt{5}-1}{\sqrt{5}+4}.E=25​+885​−2​=5​+445​−1​.

  4. Rationalize the denominator

    E=45−15+4⋅5−45−4.E=\frac{4\sqrt{5}-1}{\sqrt{5}+4}\cdot \frac{\sqrt{5}-4}{\sqrt{5}-4}.E=5​+445​−1​⋅5​−45​−4​.

    Denominator: (5+4)(5−4)=5−16=−11. (\sqrt{5}+4)(\sqrt{5}-4)=5-16=-11.(5​+4)(5​−4)=5−16=−11.

    Numerator: (45−1)(5−4)=20−165−5+4=24−175. (4\sqrt{5}-1)(\sqrt{5}-4)=20-16\sqrt{5}-\sqrt{5}+4=24-17\sqrt{5}.(45​−1)(5​−4)=20−165​−5​+4=24−175​.

    Hence, E=24−175−11=175−2411.E=\frac{24-17\sqrt{5}}{-11}=\frac{17\sqrt{5}-24}{11}.E=−1124−175​​=11175​−24​.

  5. Match with the required form

    Comparing with a5−bc,\frac{a\sqrt{5}-b}{c},ca5​−b​, we get a=17,b=24,c=11.a=17, \quad b=24, \quad c=11.a=17,b=24,c=11.

    Therefore, a+b+c=17+24+11=52.a+b+c=17+24+11=52.a+b+c=17+24+11=52.

  6. Check with options

    525252 corresponds to Option B.

  7. Comparison with stored answer

    Stored correct answer: B

    Derived answer: B

    So they agree.

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