Advertisements
Advertisements
Question
Find the value of tan22°30′. `["Hint:" "Let" θ = 45°, "use" tan theta/2 = (sin theta/2)/(cos theta/2) = (2sin theta/2 cos theta/2)/(2cos^2 theta/2) = sintheta/(1 + costheta)]`
Advertisements
Solution
Let 22°30′ = `theta/2`
∴ θ = 45°
tan22°30′ = `tan theta/2`
= `(sin theta/2)/(cos theta/2)`
= `(2sin theta/2 cos theta/2)/(2cos^2 theta/2)`
= `sintheta/(1 + costheta)`
Put θ = 45°
∴ `sintheta/(1 + costheta) = sin 45^circ/(1 + cos 45^circ)`
= `(1/sqrt(2))/(1 + 1/sqrt(2))`
= `1/(sqrt(2) + 1)`
= `(1 xx (sqrt(2) - 1))/((sqrt(2) + 1)(sqrt(2) - 1))`
= `sqrt(2) - 1`
Hence, tan22°30' = `sqrt(2) - 1`.
APPEARS IN
RELATED QUESTIONS
Prove that:
Show that :
Show that :
Prove that:
cos 20° cos 40° cos 80° = \[\frac{1}{8}\]
Prove that:
tan 20° tan 40° tan 60° tan 80° = 3
Prove that tan 20° tan 30° tan 40° tan 80° = 1.
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
sin 20° sin 40° sin 60° sin 80° = \[\frac{3}{16}\]
Show that:
sin (B − C) cos (A − D) + sin (C − A) cos (B − D) + sin (A − B) cos (C − D) = 0
If α + β = \[\frac{\pi}{2}\], show that the maximum value of cos α cos β is \[\frac{1}{2}\].
Express each of the following as the product of sines and cosines:
sin 12x + sin 4x
Express each of the following as the product of sines and cosines:
sin 2x + cos 4x
Prove that:
sin 38° + sin 22° = sin 82°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
Prove that:
Prove that:
cos 20° cos 100° + cos 100° cos 140° − 140° cos 200° = −\[\frac{3}{4}\]
Prove that:
Prove that:
Prove that:
Prove that:
If y sin ϕ = x sin (2θ + ϕ), prove that (x + y) cot (θ + ϕ) = (y − x) cot θ.
If \[m \sin\theta = n \sin\left( \theta + 2\alpha \right)\], prove that \[\tan\left( \theta + \alpha \right) \cot\alpha = \frac{m + n}{m - n}\]
Write the value of the expression \[\frac{1 - 4 \sin 10^\circ \sin 70^\circ}{2 \sin 10^\circ}\]
The value of sin 78° − sin 66° − sin 42° + sin 60° is ______.
If sin α + sin β = a and cos α − cos β = b, then tan \[\frac{\alpha - \beta}{2}\]=
If sin x + sin y = \[\sqrt{3}\] (cos y − cos x), then sin 3x + sin 3y =
Express the following as the sum or difference of sine or cosine:
`cos (7"A")/3 sin (5"A")/3`
Express the following as the product of sine and cosine.
sin 6θ – sin 2θ
Prove that:
tan 20° tan 40° tan 80° = `sqrt3`.
Prove that:
`(cos 2"A" - cos 3"A")/(sin "2A" + sin "3A") = tan "A"/2`
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
If sin(y + z – x), sin(z + x – y), sin(x + y – z) are in A.P, then prove that tan x, tan y and tan z are in A.P.
