Advertisements
Advertisements
Question
If angle θ is divided into two parts such that the tangent of one part is k times the tangent of other, and Φ is their difference, then show that sin θ = `(k + 1)/(k - 1)` sin Φ
Advertisements
Solution
Let θ = α + β.
Then tan α = k tan β
or `tanalpha/tanbeta = k/1`
Applying componendo and dividendo, we have
`(tan alpha + tan beta)/(tan alpha - tan beta) = (k + 1)/(k - 1)`
or `(sin alpha cos beta + cos alpha sin beta)/(sin alpha cos beta - cos alpha sin beta) = (k + 1)/(k - 1)`
i.e., `(sin(alpha + beta))/(sin(alpha - beta)) = (k + 1)/(k - 1)`
Given that α – β = Φ and α + β = θ.
Therefore, `sin θ/sin phi = (k + 1)/(k - 1)`
or sin θ = `(k + 1)/(k - 1) sin phi`
APPEARS IN
RELATED QUESTIONS
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:
cos2 2x – cos2 6x = sin 4x sin 8x
Prove the following:
`(sin x - siny)/(cos x + cos y)= tan (x -y)/2`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
Prove that: sin x + sin 3x + sin 5x + sin 7x = 4 cos x cos 2x sin 4x
Prove that: `((sin 7x + sin 5x) + (sin 9x + sin 3x))/((cos 7x + cos 5x) + (cos 9x + cos 3x)) = tan 6x`
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 \[\sin A = \frac{1}{2}, \cos B = \frac{12}{13}\], where \[\frac{\pi}{2}\]< A < π and \[\frac{3\pi}{2}\] < B < 2π, find tan (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:
sin (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:
\[\frac{7\pi}{12} + \cos\frac{\pi}{12} = \sin\frac{5\pi}{12} - \sin\frac{\pi}{12}\]
Prove that
Prove that:
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
If \[\tan A = \frac{m}{m - 1}\text{ and }\tan B = \frac{1}{2m - 1}\], then prove that \[A - B = \frac{\pi}{4}\].
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
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
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
Prove that:
\[\frac{1}{\sin \left( x - a \right) \sin \left( x - b \right)} = \frac{\cot \left( x - a \right) - \cot \left( x - b \right)}{\sin \left( a - b \right)}\]
If sin α sin β − cos α cos β + 1 = 0, prove that 1 + cot α tan β = 0.
Find the maximum and minimum values of each of the following trigonometrical expression:
12 sin x − 5 cos x
If α + β − γ = π and sin2 α +sin2 β − sin2 γ = λ sin α sin β cos γ, then write the value of λ.
If x cos θ = y cos \[\left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)\]then write the value of \[\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\]
If 12 sin x − 9sin2 x attains its maximum value at x = α, then write the value of sin α.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
If in ∆ABC, tan A + tan B + tan C = 6, then cot A cot B cot C =
If tan θ1 tan θ2 = k, then \[\frac{\cos \left( \theta_1 - \theta_2 \right)}{\cos \left( \theta_1 + \theta_2 \right)} =\]
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
Express the following as the sum or difference of sines and cosines:
2 sin 3x cos x
If sinθ + cosθ = 1, then find the general value of θ.
The value of tan3A - tan2A - tanA is equal to ______.
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanA = `1/2`, tanB = `1/3`, then tan(2A + B) is equal to ______.
If sinθ + cosθ = 1, then the value of sin2θ is equal to ______.
3(sinx – cosx)4 + 6(sinx + cosx)2 + 4(sin6x + cos6x) = ______.
State whether the statement is True or False? Also give justification.
If tan(π cosθ) = cot(π sinθ), then `cos(theta - pi/4) = +- 1/(2sqrt(2))`.
