Advertisements
Advertisements
प्रश्न
Write the value of \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] .
Advertisements
उत्तर
For any x ∈ [−1, 1], cos−1x represents an angle in [0, \[\pi]\] whose cosine is x.
∴ \[\cos^{- 1} \left( - \frac{1}{2} \right)\] =any angle in [0, \[\pi\]] whose cosine is \[- \frac{1}{2}\] .
\[\Rightarrow \cos^{- 1} \left( - \frac{1}{2} \right) = \frac{2\pi}{3}\]
Similarly,
\[\sin^{- 1} \left( \frac{1}{2} \right)\] = an angle in \[\left[ - \frac{\pi}{2}, \frac{\pi}{2} \right]\] whose sine is \[\frac{1}{2}\] .
\[\Rightarrow \sin^{- 1} \left( \frac{1}{2} \right) = \frac{\pi}{6}\]
∴ \[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right)\] =
\[\frac{2\pi}{3} + 2\left( \frac{\pi}{6} \right) = \frac{4\pi + 2\pi}{6} = \pi\]
Hence,
\[\cos^{- 1} \left( - \frac{1}{2} \right) + 2 \sin^{- 1} \left( \frac{1}{2} \right) = \pi\] .
APPEARS IN
संबंधित प्रश्न
Show that:
`2 sin^-1 (3/5)-tan^-1 (17/31)=pi/4`
`sin^-1(sin3)`
`sin^-1(sin4)`
Evaluate the following:
`sec^-1(sec (25pi)/6)`
Write the following in the simplest form:
`tan^-1(x/(a+sqrt(a^2-x^2))),-a<x<a`
Write the following in the simplest form:
`sin^-1{(x+sqrt(1-x^2))/sqrt2},-1<x<1`
Prove the following result
`tan(cos^-1 4/5+tan^-1 2/3)=17/6`
Evaluate:
`sec{cot^-1(-5/12)}`
Prove the following result:
`tan^-1 1/7+tan^-1 1/13=tan^-1 2/9`
Prove the following result:
`tan^-1 1/4+tan^-1 2/9=sin^-1 1/sqrt5`
Solve the following equation for x:
tan−1(x −1) + tan−1x tan−1(x + 1) = tan−13x
Solve the following equation for x:
tan−1(x + 2) + tan−1(x − 2) = tan−1 `(8/79)`, x > 0
Solve the following equation for x:
`tan^-1((x-2)/(x-4))+tan^-1((x+2)/(x+4))=pi/4`
Sum the following series:
`tan^-1 1/3+tan^-1 2/9+tan^-1 4/33+...+tan^-1 (2^(n-1))/(1+2^(2n-1))`
`sin^-1 63/65=sin^-1 5/13+cos^-1 3/5`
Evaluate the following:
`tan{2tan^-1 1/5-pi/4}`
`4tan^-1 1/5-tan^-1 1/239=pi/4`
Show that `2tan^-1x+sin^-1 (2x)/(1+x^2)` is constant for x ≥ 1, find that constant.
Write the value of cos−1 \[\left( \tan\frac{3\pi}{4} \right)\]
Write the value of cos2 \[\left( \frac{1}{2} \cos^{- 1} \frac{3}{5} \right)\]
If tan−1 x + tan−1 y = `pi/4`, then write the value of x + y + xy.
Write the value of sin \[\left\{ \frac{\pi}{3} - \sin^{- 1} \left( - \frac{1}{2} \right) \right\}\]
If \[\sin^{- 1} \left( \frac{1}{3} \right) + \cos^{- 1} x = \frac{\pi}{2},\] then find x.
Write the principal value of `tan^-1sqrt3+cot^-1sqrt3`
Wnte the value of the expression \[\tan\left( \frac{\sin^{- 1} x + \cos^{- 1} x}{2} \right), \text { when } x = \frac{\sqrt{3}}{2}\]
\[\text{ If } u = \cot^{- 1} \sqrt{\tan \theta} - \tan^{- 1} \sqrt{\tan \theta}\text{ then }, \tan\left( \frac{\pi}{4} - \frac{u}{2} \right) =\]
The value of \[\sin^{- 1} \left( \cos\frac{33\pi}{5} \right)\] is
tanx is periodic with period ____________.
