Advertisements
Advertisements
प्रश्न
Find the maximum and minimum values of each of the following trigonometrical expression:
sin x − cos x + 1
Advertisements
उत्तर
\[\text{ Let } f\left( x \right) = \sin x - \cos x + 1\]
\[\text{ We know that }\]
\[ - \sqrt{1^2 + ( - 1 )^2} \leq \sin x - \cos x \leq \sqrt{1^2 + ( - 1 )^2} for all x\]
\[ \Rightarrow - \sqrt{2} \leq \sin x - \cos x \leq \sqrt{2}\]
\[ \Rightarrow - \sqrt{2} + 1 \leq \sin x - \cos x + 1 \leq \sqrt{2} + 1\]
\[\text{ Hence maximum and minimum values of f(x) are } 1 + \sqrt{2} \text{ and } 1 - \sqrt{2} , \text{ respectively } .\]
APPEARS IN
संबंधित प्रश्न
Prove the following:
`(cos (pi + x) cos (-x))/(sin(pi - x) cos (pi/2 + x)) = cot^2 x`
Prove the following:
sin (n + 1)x sin (n + 2)x + cos (n + 1)x cos (n + 2)x = cos x
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
`(sin 5x + sin 3x)/(cos 5x + cos 3x) = tan 4x`
Prove the following:
`(sin x + sin 3x)/(cos x + cos 3x) = tan 2x`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove the following:
cot x cot 2x – cot 2x cot 3x – cot 3x cot x = 1
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
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 \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of 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 \[\sin A = \frac{1}{2}, \cos B = \frac{12}{13}\], where \[\frac{\pi}{2}\]< A < π and \[\frac{3\pi}{2}\] < B < 2π, 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)
Evaluate the following:
cos 47° cos 13° − sin 47° sin 13°
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)
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:
tan (A + B)
Prove that
\[\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin \left( A + B \right)}{\sin \left( A - B \right)}\]
Prove that:
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
Prove that:
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Write the maximum value of 12 sin x − 9 sin2 x.
tan 3A − tan 2A − tan A =
If A + B + C = π, then \[\frac{\tan A + \tan B + \tan C}{\tan A \tan B \tan C}\] is equal to
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
If \[\tan\alpha = \frac{x}{x + 1}\] and \[\tan\alpha = \frac{x}{x + 1}\], then \[\alpha + \beta\] is equal to
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
If sinθ + cosθ = 1, then find the general value of θ.
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
