Question

5sin+3cos=4
Find 3sin-5cos

Answer 1

Answer:

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Answer 2

ANSWER:-

Given:

5sin + 3cos= 4

To find:

Find the 3sin -5cos.

Solution:

$5sin \theta + 3cos \theta = 4 \\ On \: squaring \: both \: sides; \\ = > (5sin \theta + 3cos \theta) {}^{2} = (4) {}^{2} \\ \\ = > 25 {sin}^{2} \theta + 9 {cos}^{2} \theta + 30sin \theta \: cos \theta = 16 \\ \\ = > 16 {sin}^{2} \theta + 9 {sin}^{2} \theta + 9 {cos}^{2} \theta + 30sin \theta \: cos \theta = 16 \\ \\ = > 9( {sin}^{2} \theta + cos {}^{2} \theta) + 30sin \theta \: cos \theta = 16(1 - {sin}^{2} \theta) \\ \\ = > 9 + 30sin \theta \: cos \theta = 16 {cos}^{2} \theta \\ \\ = > 16 {cos}^{2} \theta - 30sin \theta \: cos \theta = 9 \\ \\ = > 16 {cos}^{2} \theta + 9 - 9 - 30sin \theta \: cos \theta = 9 \\ \\ = > 16 {cos}^{2} \theta + 9( {sin}^{2} \theta + {cos}^{2} \theta) - 30sin \theta \:cos \theta = 18 \\ \\ = > 25 {cos}^{2} \theta + 9 {sin}^{2} \theta - 30sin \theta \: cos \theta = 18 \\ \\ = > 9 {sin}^{2} \theta + 25 {cos}^{2} \theta - 30sin \theta \: cos \theta = 18 \\ \\ = > (3sin \theta - 5cos \theta) { }^{2} = 18 \\ \\ = > 3sin \theta - 5cos \theta = ± \sqrt{18} =± 3 \sqrt{2}$

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