Advertisements
Advertisements
Question
If 3 sin A + 5 cos A = 5, then show that 5 sin A – 3 cos A = ± 3
Advertisements
Solution
3 sin A + 5 cos A = 5 ....[Given]
∴ (3 sin A + 5 cos A)2 = 25 ......[Squaring both the sides]
∴ 9 sin2A + 30 sin A cos A + 25 cos2A = 25
∴ 9(1 – cos2A) + 30 sin A cos A + 25(1 – sin2A) = 25
∴ 9 – 9 cos2A + 30 sin A cos A + 25 – 25 sin2A = 25
∴ 25 sin2A – 30 sin A cos A + 9 cos2A = 9
∴ (5 sin A – 3 cos A)2 = 9 ......[∵ a2 – 2ab + b2 = (a – b)2]
∴ 5 sin A – 3 cos A = ± 3 .....[Taking square root of both sides]
APPEARS IN
RELATED QUESTIONS
Prove the following trigonometric identities:
`(\text{i})\text{ }\frac{\sin \theta }{1-\cos \theta }=\text{cosec}\theta+\cot \theta `
Prove the following identities:
`1/(sinA + cosA) + 1/(sinA - cosA) = (2sinA)/(1 - 2cos^2A)`
`(cos^3 θ + sin^3 θ)/(cos θ + sin θ) + (cos ^3 θ - sin^3 θ)/(cos θ - sin θ) = 2`
Write the value of `( 1- sin ^2 theta ) sec^2 theta.`
If `cos theta = 2/3 , " write the value of" (4+4 tan^2 theta).`
If `cos theta = 7/25 , "write the value of" ( tan theta + cot theta).`
Write True' or False' and justify your answer the following :
The value of the expression \[\sin {80}^° - \cos {80}^°\]
\[\frac{1 - \sin \theta}{\cos \theta}\] is equal to
If x = r sin θ cos ϕ, y = r sin θ sin ϕ and z = r cos θ, then
If cos A + cos2 A = 1, then sin2 A + sin4 A =
Prove the following identities:
`(tan"A"+tan"B")/(cot"A"+cot"B")=tan"A"tan"B"`
Prove that:
tan (55° + x) = cot (35° – x)
If A = 60°, B = 30° verify that tan( A - B) = `(tan A - tan B)/(1 + tan A. tan B)`.
Prove that `costheta/(1 + sintheta) = (1 - sintheta)/(costheta)`
Prove that sin4A – cos4A = 1 – 2cos2A
Prove that 2(sin6A + cos6A) – 3(sin4A + cos4A) + 1 = 0
Show that tan 7° × tan 23° × tan 60° × tan 67° × tan 83° = `sqrt(3)`
If tan θ = `x/y`, then cos θ is equal to ______.
Prove that `(1 + tan^2 A)/(1 + cot^2 A)` = sec2 A – 1
Prove that (sec θ + tan θ) (1 – sin θ) = cos θ
