Advertisements
Advertisements
Question
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
Advertisements
Solution
\[\text{ Let } f\left( x \right) = 12\sin x - 9 \sin^2 x \]
\[ = - \left( 9 \sin^2 x - 12 \sin x \right) \]
\[ = - \left[ \left( 3\sin x \right)^2 - 2 . 3 \sin x . 2 + 2^2 - 4 \right]\]
\[ = - \left[ \left( 3 \sin x - 2 \right)^2 - 4 \right]\]
\[ = 4 - \left( 3 \sin x - 2 \right)^2 \]
\[\text{ Minimum value of } \left( 3 \sin x - 2 \right)^2 \text{ is } 0 . \]
\[\text{ Therefore, maximum value of f }\left( x \right) = 4 - \left( 3 \sin x - 2 \right)^2 \text{ is } 4 . \]
\[\text{ We are given that } 12\sin x - 9 \sin^2 x \text{ will attain its maximum value at } x = \alpha . \]
\[ \therefore 12\sin\alpha - 9 \sin^2 \alpha = 4\]
\[ \Rightarrow - 9 \sin^2 \alpha + 12\sin\alpha - 4 = 0\]
\[ \Rightarrow 9 \sin^2 \alpha - 12 \sin\alpha + 4 = 0\]
\[ \Rightarrow 9 \sin^2 \alpha - 6\sin\alpha - 6\sin\alpha + 4 = 0\]
\[ \Rightarrow 3\sin\alpha\left( 3\sin\alpha - 2 \right) - 2\left( 3\sin\alpha - 2 \right) = 0\]
\[ \Rightarrow \left( 3\sin\alpha - 2 \right)\left( 3\sin\alpha - 2 \right) = 0\]
\[ \therefore \sin\alpha = \frac{2}{3}\]
APPEARS IN
RELATED QUESTIONS
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Find the value of: sin 75°
Find the value of: tan 15°
Prove the following:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
`(sin x - sin 3x)/(sin^2 x - cos^2 x) = 2sin x`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
If \[\sin A = \frac{12}{13}\text{ and } \sin B = \frac{4}{5}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
cos (A + B)
If \[\cos A = - \frac{24}{25}\text{ and }\cos B = \frac{3}{5}\], where π < A < \[\frac{3\pi}{2}\text{ and }\frac{3\pi}{2}\]< B < 2π, find the following:
cos (A + B)
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that
Prove that
If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If α, β are two different values of x lying between 0 and 2π, which satisfy the equation 6 cos x + 8 sin x = 9, find the value of sin (α + β).
Prove that:
If angle \[\theta\] is divided into two parts such that the tangents of one part is \[\lambda\] times the tangent of other, and \[\phi\] is their difference, then show that\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Show that sin 100° − sin 10° is positive.
Write the maximum value of 12 sin x − 9 sin2 x.
Write the interval in which the value of 5 cos x + 3 cos \[\left( x + \frac{\pi}{3} \right) + 3\] lies.
tan 3A − tan 2A − tan A =
If cot (α + β) = 0, sin (α + 2β) is equal to
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
If tan (A − B) = 1 and sec (A + B) = \[\frac{2}{\sqrt{3}}\], the smallest positive value of B is
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
If sinθ + cosθ = 1, then find the general value of θ.
If cotθ + tanθ = 2cosecθ, then find the general value of θ.
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If sinx + cosx = a, then sin6x + cos6x = ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
State whether the statement is True or False? Also give justification.
If cosecx = 1 + cotx then x = 2nπ, 2nπ + `pi/2`
