Advertisements
Advertisements
Question
Prove that \[\left( 2\sqrt{3} + 3 \right) \sin x + 2\sqrt{3} \cos x\] lies between \[- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right)\]
Advertisements
Solution
\[\text{ Let } f\left( x \right) = (2\sqrt{3} + 3) \sin x + 2\sqrt{3}\cos x\]
\[\text{ We know that }, \]
\[ - \sqrt{\left( 2\sqrt{3} + 3 \right)^2 + \left( 2\sqrt{3} \right)^2} \leq f\left( x \right) \leq \sqrt{\left( 2\sqrt{3} + 3 \right)^2 + \left( 2\sqrt{3} \right)^2}\]
\[ \Rightarrow - \sqrt{12 + 9 + 12\sqrt{3} + 12} \leq f\left( x \right) \leq \sqrt{12 + 9 + 12\sqrt{3} + 12}\]
\[ \Rightarrow - \sqrt{33 + 12\sqrt{3}} \leq f\left( x \right) \leq \sqrt{33 + 12\sqrt{3}}\]
\[\text{ Disclaimer } : \text{ Instead of }- \left( 2\sqrt{3} + \sqrt{15} \right) \text{ and } \left( 2\sqrt{3} + \sqrt{15} \right),\text{ it should be } - \sqrt{33 + 12\sqrt{3}}\text{ and } \sqrt{33 + 12\sqrt{3}} .\]
APPEARS IN
RELATED QUESTIONS
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
`cos ((3pi)/ 2 + x ) cos(2pi + x) [cot ((3pi)/2 - x) + cot (2pi + x)]= 1`
Prove the following:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
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:
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)
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], where A lies in the second quadrant and B in the third quadrant, find the values of the following:
cos (A + B)
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
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:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
If tan (A + B) = x and tan (A − B) = y, find the values of tan 2A and tan 2B.
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).
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
If A + B = C, then write the value of tan A tan B tan C.
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
tan 3A − tan 2A − tan A =
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
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 \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
The maximum distance of a point on the graph of the function y = `sqrt(3)` sinx + cosx from x-axis is ______.
