Advertisements
Advertisements
Question
If secx cos5x + 1 = 0, where 0 < x ≤ `pi/2`, then find the value of x.
Advertisements
Solution
secx cos5x = –1
⇒ cos5x = `(-1)/secx`
We know that
secx = `1/cosx`
⇒ cos5x + cosx = 0
By transformation formula of T-ratios,
We know that
cosA + cosB = `2cos(("A" + "B")/2) cos(("A" - "B")/2)`
⇒ `2cos((5x + x)/2) cos((5x - x)/2)` = 0
⇒ 2cos3x cos2x = 0
⇒ cos3x = 0 or cos2x = 0
∵ 0 < x ≤ `pi/2`
Therefore, 0 < 2x ≤ π or 0 < 3x ≤ `(3pi)/2`
Therefore, 2x = `pi/2`
⇒ x = `pi/4`
3x = `pi/2`
⇒ x = `pi/6`
Or 3x = `(3pi)/2`
⇒ x = `pi/2`
Hence, x = `pi/6, pi/4, pi/2`.
APPEARS IN
RELATED QUESTIONS
Prove that:
Prove that:
cos 10° cos 30° cos 50° cos 70° = \[\frac{3}{16}\]
Prove that:
sin 10° sin 50° sin 60° sin 70° = \[\frac{\sqrt{3}}{16}\]
Prove that:
\[\tan x \tan \left( \frac{\pi}{3} - x \right) \tan \left( \frac{\pi}{3} + x \right) = \tan 3x\]
Express each of the following as the product of sines and cosines:
cos 12x - cos 4x
Prove that:
cos 100° + cos 20° = cos 40°
Prove that:
sin 40° + sin 20° = cos 10°
Prove that:
sin 50° − sin 70° + sin 10° = 0
Prove that:
cos 80° + cos 40° − cos 20° = 0
Prove that:
cos 20° + cos 100° + cos 140° = 0
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
Prove that:
If cosec A + sec A = cosec B + sec B, prove that tan A tan B = \[\cot\frac{A + B}{2}\].
Write the value of \[\sin\frac{\pi}{15}\sin\frac{4\pi}{15}\sin\frac{3\pi}{10}\]
Write the value of \[\frac{\sin A + \sin 3A}{\cos A + \cos 3A}\]
If sin 2 θ + sin 2 ϕ = \[\frac{1}{2}\] and cos 2 θ + cos 2 ϕ = \[\frac{3}{2}\], then cos2 (θ − ϕ) =
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 \[\frac{\alpha - \beta}{2}\]=
sin 47° + sin 61° − sin 11° − sin 25° is equal to
If A, B, C are in A.P., then \[\frac{\sin A - \sin C}{\cos C - \cos A}\]=
Express the following as the sum or difference of sine or cosine:
`sin "A"/8 sin (3"A")/8`
Express the following as the product of sine and cosine.
cos 2θ – cos θ
Prove that:
2 cos `pi/13` cos \[\frac{9\pi}{13} + \text{cos} \frac{3\pi}{13} + \text{cos} \frac{5\pi}{13}\] = 0
Prove that cos 20° cos 40° cos 60° cos 80° = `3/16`.
Evaluate-
cos 20° + cos 100° + cos 140°
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)]`
