Advertisements
Advertisements
Question
If 3 sin θ + 5 cos θ = 5, prove that 5 sin θ – 3 cos θ = ± 3.
Advertisements
Solution
Given 3 sin θ + 5 cos θ = 5
Squaring on both sides for both the equations
⇒ 9 sin2θ + 25 cos2θ + 30 sinθ cosθ = 25
⇒ 25 sin2θ + 9 cos2θ − 30 sinθ cosθ = x2
Adding the equations;
⇒ 34 (sin2θ + cos2θ) = 25 + x2
⇒ x2 = 34 − 25 = 9
⇒ x = ±3
∴ 5 sinθ − 3 cosθ = ±3
Hence proved.
APPEARS IN
RELATED QUESTIONS
If `sec alpha=2/sqrt3` , then find the value of `(1-cosecalpha)/(1+cosecalpha)` where α is in IV quadrant.
If `x/a=y/b = z/c` show that `x^3/a^3 + y^3/b^3 + z^3/c^3 = (3xyz)/(abc)`.
Prove the following trigonometric identities.
`(1 + cos A)/sin^2 A = 1/(1 - cos A)`
Prove the following identities:
`(sec A - 1)/(sec A + 1) = (1 - cos A)/(1 + cos A)`
`cosec theta (1+costheta)(cosectheta - cot theta )=1`
`(1+tan^2theta)(1+cot^2 theta)=1/((sin^2 theta- sin^4theta))`
Prove the following identity :
`(cot^2θ(secθ - 1))/((1 + sinθ)) = sec^2θ((1-sinθ)/(1 + secθ))`
If `asin^2θ + bcos^2θ = c and p sin^2θ + qcos^2θ = r` , prove that (b - c)(r - p) = (c - a)(q - r)
Without using trigonometric table , evaluate :
`(sin47^circ/cos43^circ)^2 - 4cos^2 45^circ + (cos43^circ/sin47^circ)^2`
If x = r sin θ cos Φ, y = r sin θ sin Φ and z = r cos θ, prove that x2 + y2 + z2 = r2.
Prove that sin2 θ + cos4 θ = cos2 θ + sin4 θ.
Prove that sin( 90° - θ ) sin θ cot θ = cos2θ.
Prove the following identities.
`sqrt((1 + sin theta)/(1 - sin theta)) + sqrt((1 - sin theta)/(1 + sin theta))` = 2 sec θ
Choose the correct alternative:
cos θ. sec θ = ?
Choose the correct alternative:
cot θ . tan θ = ?
If tan θ = `7/24`, then to find value of cos θ complete the activity given below.
Activity:
sec2θ = 1 + `square` ......[Fundamental tri. identity]
sec2θ = 1 + `square^2`
sec2θ = 1 + `square/576`
sec2θ = `square/576`
sec θ = `square`
cos θ = `square` .......`[cos theta = 1/sectheta]`
Prove that `(1 + sin "B")/"cos B" + "cos B"/(1 + sin "B")` = 2 sec B
Prove the following:
`1 + (cot^2 alpha)/(1 + "cosec" alpha)` = cosec α
If tan θ + sec θ = l, then prove that sec θ = `(l^2 + 1)/(2l)`.
(1 + sin A)(1 – sin A) is equal to ______.
