Question

If `(3 sin theta + 5 cos theta ) = 5, ` prove that ` (5 sin theta - 3 cos theta ) = pm 3. `

Answer 1

We have
`(3 sin theta + 5 cos theta)^(2) + (5 sin theta - 3 cos theta)^(2) `
` = 9 (sin^(2) theta + cos^(2) theta ) + 25 ( sin^(2) theta + cos^(2) theta) `
` = (9+ 25)= 34.`
` therefore (3 sin theta + 5 cos theta)^(2) + ( 5 sin theta - 3 cos theta)^(2) =34 `
` rArr 5^(2) + (5 sin theta - 3 cos theta)^(2) = 34 " " [ because 3 sin theta + 5 cos theta = 5 ] `
` rArr (5 sin theta - 3 cos theta) = pm 3 " " `[ taking square root on each side]
Hence, `(5 sin theta - 3 cos theta) = pm 3. `