Advertisements
Advertisements
प्रश्न
If sin (π cos x) = cos (π sin x), then sin 2x = ______.
पर्याय
- \[\pm \frac{3}{4}\]
- \[\pm \frac{4}{3}\]
- \[\pm \frac{1}{3}\]
None of these
Advertisements
उत्तर
If sin (π cos x) = cos (π sin x), then sin 2x = `underlinebb(+- 3/4)`.
Explanation:
sin (π cos x) = cos (π sin x)
∵ sin θ = cos (90 - θ)
`=> cos (pi/2 - pi cos x) = cos (pi sin x)`
`=> pi/2 - pi cos x = pi sin x`
`=> pi/2 = pi (sin x + cos x)`
`=> cancel(pi)/2 = cancel(pi) (sin x + cos x)`
`=> sin x + cos x = 1/2`
Squaring on both sides,
`=> (sin x + cos x)^2 = 1/4`
`=> sin^2x + cos^2x + 2 sin x * cos x = 1/4`
`=> 1 + 2 sin x * cos x = 1/4 ...[sin^2x + cos^2x = 1]`
`=> sin 2x = 1/4 - 1`
⇒ sin 2x = `(- 3)/4`
APPEARS IN
संबंधित प्रश्न
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
cos2 2x – cos2 6x = sin 4x sin 8x
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`
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/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:
cos (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:
sin (A + B)
Prove that:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that
Prove that
Prove that:
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If sin α + sin β = a and cos α + cos β = b, show that
Prove that:
Prove that:
If tan α = x +1, tan β = x − 1, show that 2 cot (α − β) = x2.
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Write the interval in which the value of 5 cos x + 3 cos \[\left( x + \frac{\pi}{3} \right) + 3\] lies.
If \[\frac{\cos \left( x - y \right)}{\cos \left( x + y \right)} = \frac{m}{n}\] then write the value of tan x tan y.
If A + B = C, then write the value of tan A tan B tan C.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
The value of \[\sin^2 \frac{5\pi}{12} - \sin^2 \frac{\pi}{12}\]
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 cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
Express the following as the sum or difference of sines and cosines:
2 cos 3x sin 2xa
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
Match each item given under column C1 to its correct answer given under column C2.
| C1 | C2 |
| (a) `(1 - cosx)/sinx` | (i) `cot^2 x/2` |
| (b) `(1 + cosx)/(1 - cosx)` | (ii) `cot x/2` |
| (c) `(1 + cosx)/sinx` | (iii) `|cos x + sin x|` |
| (d) `sqrt(1 + sin 2x)` | (iv) `tan x/2` |
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
State whether the statement is True or False? Also give justification.
If tanA = `(1 - cos B)/sinB`, then tan2A = tanB
In the following match each item given under the column C1 to its correct answer given under the column C2:
| Column A | Column B |
| (a) sin(x + y) sin(x – y) | (i) cos2x – sin2y |
| (b) cos (x + y) cos (x – y) | (ii) `(1 - tan theta)/(1 + tan theta)` |
| (c) `cot(pi/4 + theta)` | (iii) `(1 + tan theta)/(1 - tan theta)` |
| (d) `tan(pi/4 + theta)` | (iv) sin2x – sin2y |
