Question

If 3 cos θ = 5 sin θ, then the value of $\frac{5sin\Theta-2sec^{3}\Theta+2cos\Theta}{5sin\Theta+2sec^{3}\Theta-2cos\Theta}$ is
(a) tex]\frac{271}{979}[/tex]
(b)$\frac{316}{2937}$
(c)$\frac{542}{2937}$
(d) None of these

Answer 1

SOLUTION :  

The correct option is (a) : 271/979

Given : 3 cos θ = 5 sin θ

sin θ/cos θ = ⅗

tan θ = ⅗  

[ sin θ/cos θ = tan θ]

In right angle ∆ ,  

tan θ =  perpendicular/base = 3/5

perpendicular = 3 , base = 5

Hypotenuse = √( perpendicular)² + (Base)²

[By Pythagoras theorem]

Hypotenuse = √ 3² + 5² = √(9 + 25) = √34

Hypotenuse = √34  

sin θ = perpendicular / hypotenuse  = 3/√34

cos θ = base/ hypotenuse = 5/√34

sec θ = 1/cos θ = 1/ (√5/√34) = √34/5

The value of : 5 sin θ - 2 sec³ θ + 2 cos θ / 5 sin θ + 2 sec³ θ - 2 cos θ

= [ 5(3/√34) - 2(√34/5)³ + 2 (5/√34)]  / [ 5(3/√34) + 2(√34/5)³ - 2 (5/√34)]

= (15/√34 - 2 × 34× √34 /125 + 10/√34) / (15/√34 + 2 × 34 × √34 /125 - 10/√34)

= (25/√34 - 68√34/125) / (5/√34 + 68√34/125)  

= [(25 × 125 - 68 √34 ×√34)/125√34] /   [(5 × 125 +  68 √34 ×√34)/125√34]

= (3125 - 68 × 34) /125√34 / (625 +  68 × 34)/125√34

= [(3125 - 2312)/125√34] / [(625 + 2312)/ 125√34]

=( 813/ 125√34) /( 2937/125√34)

= ( 813 / 125√34) ×  (125√34 /2937)

= 813/2937

= 271/979  

5 sin θ - 2 sec³ θ + 2 cos θ / 5 sin θ + 2 sec³ θ - 2 cos θ = 271/979

Hence, the value of 5 sin θ - 2 sec³ θ + 2 cos θ / 5 sin θ + 2 sec³ θ - 2 cos θ is 271/979 .

HOPE THIS ANSWER WILL HELP YOU..

Answer 2

Step-by-step explanation:

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