Educational

If 12 sin x − 9 sin²x Attains Maximum Value at x = α, Find sin α If 12 sin x − 9 sin²x Attains Maximum Value at x = α, Find sin α Question: If \[ 12\sin x-9\sin^2x \] attains its maximum value at \[ x=\alpha \] then write the value of \[ \sin\alpha […]

Find the Maximum Value of 12 sin x − 9 sin²x Find the Maximum Value of 12 sin x − 9 sin²x Question: \[ 12\sin x-9\sin^2x \] Write its maximum value. Solution Let \[ \sin x=t \] where \[ -1\le t\le1 \] Then, \[ 12\sin x-9\sin^2x = 12t-9t^2 \] \[ = -9\left(t^2-\frac43 t\right) \] \[

Find the Maximum and Minimum Values of 3 cos x + 4 sin x + 5 Find the Maximum and Minimum Values of 3 cos x + 4 sin x + 5 Question: \[ 3\cos x+4\sin x+5 \] Write its maximum and minimum values. Solution \[ a\cos x+b\sin x \] has maximum value \[ \sqrt{a^2+b^2}

If x cos θ = y cos(θ + 2π/3) = z cos(θ + 4π/3), Find 1/x + 1/y + 1/z If x cos θ = y cos(θ + 2π/3) = z cos(θ + 4π/3), Find 1/x + 1/y + 1/z Question: \[ x\cos\theta = y\cos\left(\theta+\frac{2\pi}{3}\right) = z\cos\left(\theta+\frac{4\pi}{3}\right) \] Find \[ \frac1x+\frac1y+\frac1z \] Solution \[ x\cos\theta

If α + β − γ = π, Find λ If α + β − γ = π, Find λ Question: \[ \alpha+\beta-\gamma=\pi \] and \[ \sin^2\alpha+\sin^2\beta-\sin^2\gamma = \lambda \sin\alpha\sin\beta\cos\gamma \] Find \(\lambda\). Solution \[ \gamma=\alpha+\beta-\pi \] \[ \cos\gamma = -\cos(\alpha+\beta) \] \[ = \sin\alpha\sin\beta-\cos\alpha\cos\beta \] \[ \sin^2\gamma = \sin^2(\alpha+\beta) \] \[ = (\sin\alpha\cos\beta+\cos\alpha\sin\beta)^2 \]

Find the Value of cos(π/12) − sin(π/2) Find the Value of cos(π/12) − sin(π/2) Question: The value of \[ \cos\left(\frac{\pi}{12}\right)-\sin\left(\frac{\pi}{2}\right) \] is ……………………………………. Solution We know that \[ \sin\left(\frac{\pi}{2}\right)=1 \] Also, \[ \cos\left(\frac{\pi}{12}\right) = \cos15^\circ \] Using the identity: \[ \cos(A-B)=\cos A\cos B+\sin A\sin B \] Take \[ A=45^\circ, \qquad B=30^\circ \] Then, \[ \cos15^\circ

Find the Value of tan 5x tan 3x tan 2x − tan 5x + tan 3x + tan 2x Find the Value of tan 5x tan 3x tan 2x − tan 5x + tan 3x + tan 2x Question: The value of \[ \tan5x\tan3x\tan2x-\tan5x+\tan3x+\tan2x \] is ……………………………………. Solution Using the identity: \[ \tan(A+B) = \frac{

If tan x = 1/2 and tan y = 1/3, Find the Value of x + y If tan x = 1/2 and tan y = 1/3, Find the Value of x + y Question: If \[ \tan x=\frac12 \] and \[ \tan y=\frac13 \] then the value of \[ x+y \] is ……………………………………. Solution

If cos²(π/6 + x) − sin²(π/6 − x) = k cos 2x, Find k If cos²(π/6 + x) − sin²(π/6 − x) = k cos 2x, Find k Question: If \[ \cos^2\left(\frac{\pi}{6}+x\right) – \sin^2\left(\frac{\pi}{6}-x\right) = k\cos2x \] then \[ k= \] ……………………………………….. Solution Using the identities: \[ \cos^2\theta=\frac{1+\cos2\theta}{2} \] and \[ \sin^2\theta=\frac{1-\cos2\theta}{2} \] Therefore, \[

If sin x cos y = 1/4 and 3 tan x = 4 tan y, Find sin(x − y) If sin x cos y = 1/4 and 3 tan x = 4 tan y, Find sin(x − y) Question: If \[ \sin x\cos y=\frac14 \] and \[ 3\tan x=4\tan y \] then \[ \sin(x-y) \]