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
Find the value of: sin 75°
Prove the following: `cos (pi/4 xx x) cos (pi/4 - y) - sin (pi/4 - x)sin (pi/4 - y) = sin (x + y)`
Prove the following:
`cos ((3pi)/4 + x) - cos((3pi)/4 - x) = -sqrt2 sin 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 that: `(cos x - cosy)^2 + (sin x - sin y)^2 = 4 sin^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 \[\sin A = \frac{4}{5}\] and \[\cos B = \frac{5}{13}\], where 0 < A, \[B < \frac{\pi}{2}\], find the value of the following:
cos (A − B)
If \[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\], 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
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that:
cos2 A + cos2 B − 2 cos A cos B cos (A + B) = sin2 (A + B)
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
Prove that:
\[\tan\frac{\pi}{12} + \tan\frac{\pi}{6} + \tan\frac{\pi}{12}\tan\frac{\pi}{6} = 1\]
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
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).
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 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\[\sin\theta = \frac{\lambda + 1}{\lambda - 1}\sin\phi\]
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.
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
The value of cos (36° − A) cos (36° + A) + cos (54° + A) cos (54° − A) is
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 tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If tanθ = `(sinalpha - cosalpha)/(sinalpha + cosalpha)`, then show that sinα + cosα = `sqrt(2)` cosθ.
[Hint: Express tanθ = `tan (alpha - pi/4) theta = alpha - pi/4`]
If sinθ + cosθ = 1, then find the general value of θ.
The value of tan 75° - cot 75° is equal to ______.
