Advertisements
Advertisements
प्रश्न
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Advertisements
उत्तर
\[\text{ Let } f\left( x \right) = 5 \cos x + 3 \sin\left( \frac{\pi}{6} - x \right) + 4\]
\[\text{ Now } f\left( x \right) = 5\cos x + 3\left( \sin30°\cos x - \cos30°\sin x \right) + 4\]
\[ = 5\cos x + \frac{3}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x + 4\]
\[ = \frac{13}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x + 4\]
\[\text{ We know that }\]
\[ - \sqrt{\left( \frac{13}{2} \right)^2 + \left( - \frac{3\sqrt{3}}{2} \right)^2} \leq \frac{13}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x \leq \sqrt{\left( \frac{13}{2} \right)^2 + \left( - \frac{3\sqrt{3}}{2} \right)^2} \text{ for all x }\]
\[\text{ Therefore }, \]
\[ - \sqrt{\frac{169 + 27}{4}} \leq \frac{13}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x \leq \sqrt{\frac{169 + 27}{4}}\]
\[ \Rightarrow - \frac{14}{2} + 4 \leq \frac{13}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x + 4 \leq \frac{14}{2} + 4\]
\[ \Rightarrow - 3 \leq \frac{13}{2}\cos x - \frac{3\sqrt{3}}{2}\sin x + 4 \leq 11\]
\[\text{ Hence, maximum and minimun values of } f\left( x \right) \text{ are 11 and - 3, respectively } .\]
APPEARS IN
संबंधित प्रश्न
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Prove that `2 sin^2 pi/6 + cosec^2 (7pi)/6 cos^2 pi/3 = 3/2`
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
Prove that: `(cos x - cosy)^2 + (sin x - sin y)^2 = 4 sin^2 (x - y)/2`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
If \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
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:
sin (A + B)
If \[\tan A = \frac{3}{4}, \cos B = \frac{9}{41}\], where π < A < \[\frac{3\pi}{2}\] and 0 < B <\[\frac{\pi}{2}\], find tan (A + B).
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)
Prove that:
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
If tan A + tan B = a and cot A + cot B = b, prove that cot (A + B) \[\frac{1}{a} - \frac{1}{b}\].
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).
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
If \[\tan\theta = \frac{\sin\alpha - \cos\alpha}{\sin\alpha + \cos\alpha}\] , then show that \[\sin\alpha + \cos\alpha = \sqrt{2}\cos\theta\].
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
If A + B = C, then write the value of tan A tan B tan C.
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
If tan (π/4 + x) + tan (π/4 − x) = a, then tan2 (π/4 + x) + tan2 (π/4 − x) =
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
If α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.
The value of tan3A - tan2A - tanA is equal to ______.
If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
State whether the statement is True or False? Also give justification.
If cosecx = 1 + cotx then x = 2nπ, 2nπ + `pi/2`
State whether the statement is True or False? Also give justification.
If tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.
