Advertisements
Advertisements
प्रश्न
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)
Advertisements
उत्तर
Given:
\[\cos A = - \frac{12}{13}\text{ and }\cot B = \frac{24}{7}\]
A lies in thesecond quadrant and B lies in the third quadrant .
We know that sine function is positive in thesecond quadrant and in thethird quadrant, both sine and cosine functions are negative.
Therefore,
\[\sin A = \sqrt{1 - \cos^2 A} = \sqrt{1 - \left( \frac{- 12}{13} \right)^2} = \sqrt{1 - \frac{144}{169}} = \sqrt{\frac{25}{169}} = \frac{5}{13}\]
\[\sin B = - \frac{1}{\sqrt{1 + \cot^2 B}} = - \frac{1}{\sqrt{1 + \left( \frac{24}{7} \right)^2}} = \frac{- 1}{\sqrt{1 + \frac{576}{49}}} = \frac{- 1}{\sqrt{\frac{625}{49}}} = \frac{- 7}{25}\]
\[\cos B = - \sqrt{1 - \sin^2 B} = - \sqrt{1 - \left( \frac{- 7}{25} \right)^2} = - \sqrt{1 - \frac{49}{625}} = - \sqrt{\frac{576}{625}} = - \frac{24}{25}\]
Now,
\[ \tan\left( A + B \right) = \frac{\sin\left( A + B \right)}{\cos\left( A + B \right)} = \frac{\frac{- 36}{325}}{\frac{323}{325}} = - \frac{36}{323}\]
APPEARS IN
संबंधित प्रश्न
Prove that: `2 sin^2 (3pi)/4 + 2 cos^2 pi/4 + 2 sec^2 pi/3 = 10`
Prove the following: `(tan(pi/4 + x))/(tan(pi/4 - x)) = ((1+ tan x)/(1- tan x))^2`
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
If \[\sin A = \frac{1}{2}, \cos B = \frac{\sqrt{3}}{2}\], where \[\frac{\pi}{2}\] < A < π and 0 < B < \[\frac{\pi}{2}\], find the following:
tan (A + B)
Evaluate the following:
sin 78° cos 18° − cos 78° sin 18°
Evaluate the following:
sin 36° cos 9° + cos 36° sin 9°
Prove that:
Prove that \[\frac{\tan 69^\circ + \tan 66^\circ}{1 - \tan 69^\circ \tan 66^\circ} = - 1\].
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
Prove that:
\[\frac{\sin \left( A - B \right)}{\cos A \cos B} + \frac{\sin \left( B - C \right)}{\cos B \cos C} + \frac{\sin \left( C - A \right)}{\cos C \cos A} = 0\]
Prove that:
\[\frac{\tan \left( A + B \right)}{\cot \left( A - B \right)} = \frac{\tan^2 A - \tan^2 B}{1 - \tan^2 A \tan^2 B}\]
Prove that:
tan 36° + tan 9° + tan 36° tan 9° = 1
Prove that:
\[\frac{\tan^2 2x - \tan^2 x}{1 - \tan^2 2x \tan^2 x} = \tan 3x \tan x\]
If tan A = x tan B, prove that
\[\frac{\sin \left( A - B \right)}{\sin \left( A + B \right)} = \frac{x - 1}{x + 1}\]
If cos A + sin B = m and sin A + cos B = n, prove that 2 sin (A + B) = m2 + n2 − 2.
If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:
Prove that:
If α and β are two solutions of the equation a tan x + b sec x = c, then find the values of sin (α + β) and cos (α + β).
Find the maximum and minimum values of each of the following trigonometrical expression:
12 cos x + 5 sin x + 4
Find the maximum and minimum values of each of the following trigonometrical expression:
\[5 \cos x + 3 \sin \left( \frac{\pi}{6} - x \right) + 4\]
Reduce each of the following expressions to the sine and cosine of a single expression:
\[\sqrt{3} \sin x - \cos x\]
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin x
Show that sin 100° − sin 10° is positive.
If a = b \[\cos \frac{2\pi}{3} = c \cos\frac{4\pi}{3}\] then write the value of ab + bc + ca.
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 \[\left( 0, \frac{\pi}{2} \right)\]
If A + B + C = π, then sec A (cos B cos C − sin B sin C) is equal to
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 sin (π cos x) = cos (π sin x), then sin 2x = ______.
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
The maximum value of \[\sin^2 \left( \frac{2\pi}{3} + x \right) + \sin^2 \left( \frac{2\pi}{3} - x \right)\] is
If tan 69° + tan 66° − tan 69° tan 66° = 2k, then k =
Express the following as the sum or difference of sines and cosines:
2 cos 7x cos 3x
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
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
If sinθ + cosecθ = 2, then sin2θ + cosec2θ is equal to ______.
If α + β = `pi/4`, then the value of (1 + tan α)(1 + tan β) is ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
Given x > 0, the values of f(x) = `-3cos sqrt(3 + x + x^2)` lie in the interval ______.
