Advertisements
Advertisements
प्रश्न
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = ______.
उत्तर
If 3 tan (θ – 15°) = tan (θ + 15°), 0° < θ < 90°, then θ = `pi/4`.
Explanation:
Given that 3 tan (θ – 15°) = tan (θ + 15°)
Which can be rewritten as `(tan(theta + 15^circ))/(tan(theta - 15^circ)) = 3/1`
Applying componendo and Dividendo
We get `(tan(theta + 15^circ) + tan(theta - 15^circ))/(tan(theta + 15^circ) - tan(theta - 15^circ))` = 2
⇒ `(sin(theta + 15^circ) cos(theta - 15^circ) + sin(theta - 15^circ) cos(theta + 15^circ))/(sin(theta + 15^circ) cos(theta - 15^circ) - sin(theta - 15^circ) cos(theta + 15^circ))` = 2
⇒ `(sin 2theta)/(sin30^circ)` = 2
i.e., sin 2θ = 1
Giving θ = `pi/4`
APPEARS IN
संबंधित प्रश्न
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Find the value of: sin 75°
Prove the following:
`(cos9x - cos5x)/(sin17x - sin 3x) = - (sin2x)/(cos 10x)`
Prove the following:
cos 4x = 1 – 8sin2 x cos2 x
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)
Evaluate the following:
cos 80° cos 20° + sin 80° sin 20°
Prove that
\[\frac{\tan A + \tan B}{\tan A - \tan B} = \frac{\sin \left( A + B \right)}{\sin \left( A - B \right)}\]
Prove that
Prove that:
Prove that:
sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
Prove that: \[\frac{\sin \left( A + B \right) + \sin \left( A - B \right)}{\cos \left( A + B \right) + \cos \left( A - B \right)} = \tan A\]
Prove that:
sin2 B = sin2 A + sin2 (A − B) − 2 sin A cos B sin (A − B)
Prove that:
tan 8x − tan 6x − tan 2x = tan 8x tan 6x tan 2x
If sin α + sin β = a and cos α + cos β = b, show that
Prove that:
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
If sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If tan \[\alpha = \frac{1}{1 + 2^{- x}}\] and \[\tan \beta = \frac{1}{1 + 2^{x + 1}}\] then write the value of α + β lying in the interval \[\left( 0, \frac{\pi}{2} \right)\]
tan 20° + tan 40° + \[\sqrt{3}\] tan 20° tan 40° is equal to
If \[\tan A = \frac{a}{a + 1}\text{ and } \tan B = \frac{1}{2a + 1}\]
tan 3A − tan 2A − tan A =
If \[\cos P = \frac{1}{7}\text{ and }\cos Q = \frac{13}{14}\], where P and Q both are acute angles. Then, the value of P − Q is
If A − B = π/4, then (1 + tan A) (1 − tan B) is equal to
Show that 2 sin2β + 4 cos (α + β) sin α sin β + cos 2(α + β) = cos 2α
If sinθ + cosθ = 1, then find the general value of θ.
If sin(θ + α) = a and sin(θ + β) = b, then prove that cos 2(α - β) - 4ab cos(α - β) = 1 - 2a2 - 2b2
[Hint: Express cos(α - β) = cos((θ + α) - (θ + β))]
If cos(θ + Φ) = m cos(θ – Φ), then prove that 1 tan θ = `(1 - m)/(1 + m) cot phi`
[Hint: Express `(cos(theta + Φ))/(cos(theta - Φ)) = m/1` and apply Componendo and Dividendo]
If tan θ = 3 and θ lies in third quadrant, then the value of sin θ ______.
