Advertisements
Advertisements
Question
Reduce each of the following expressions to the sine and cosine of a single expression:
24 cos x + 7 sin x
Advertisements
Solution
\[\text{ Let } f(x) = 24 \cos x + 7\sin x\]
\[\text{ Dividing and multiplying by } \sqrt{{24}^2 + 7^2}, i . e . \text{ by 25, we get }: \]
\[ f(x) = 25\left( \frac{24}{25} \cos x + \frac{7}{25}\sin x \right)\]
\[ \Rightarrow f(x) = 25(\sin\alpha \cos x + \cos\alpha \sin x), \text{ where } \sin\alpha = \frac{24}{25} and \cos\alpha = \frac{7}{25}\]
\[ \Rightarrow f(x) = 25 \sin(\alpha + x), \text{ where } \tan\alpha = \frac{24}{7} . \]
\[\text{ Again }, \]
\[ f(x) = 25\left( \frac{24}{25} \cos x + \frac{7}{25}\sin x \right)\]
\[ \Rightarrow f(x) = 25(\cos\alpha \cos x + \sin\alpha \sin x), \text{ where } \cos\alpha = \frac{24}{25}, \sin\alpha = \frac{7}{25} . \]
\[ \Rightarrow f(x) = 25 \cos(\alpha - x), \text{ where }\tan\alpha = \frac{7}{24} .\]
APPEARS IN
RELATED QUESTIONS
Prove that: `sin^2 pi/6 + cos^2 pi/3 - tan^2 pi/4 = -1/2`
Find the value of: tan 15°
Prove the following:
`(cos 4x + cos 3x + cos 2x)/(sin 4x + sin 3x + sin 2x) = cot 3x`
Prove the following:
cos 6x = 32 cos6 x – 48 cos4 x + 18 cos2 x – 1
Prove that: `(cos x + cos y)^2 + (sin x - sin y )^2 = 4 cos^2 (x + y)/2`
Prove that: sin 3x + sin 2x – sin x = 4sin x `cos x/2 cos (3x)/2`
If \[\sin A = \frac{3}{5}, \cos B = - \frac{12}{13}\], where A and B both lie in second quadrant, find the value of sin (A + B).
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:
cos 80° cos 20° + sin 80° sin 20°
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)
Prove that
Prove that:
Prove that:
If \[\tan A = \frac{5}{6}\text{ and }\tan B = \frac{1}{11}\], prove that \[A + B = \frac{\pi}{4}\].
Prove that:
\[\cos^2 45^\circ - \sin^2 15^\circ = \frac{\sqrt{3}}{4}\]
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 sin2 (n + 1) A − sin2 nA = sin (2n + 1) A sin A.
If x lies in the first quadrant and \[\cos x = \frac{8}{17}\], then prove that:
If sin (α + β) = 1 and sin (α − β) \[= \frac{1}{2}\], where 0 ≤ α, \[\beta \leq \frac{\pi}{2}\], then find the values of tan (α + 2β) and tan (2α + β).
If sin α + sin β = a and cos α + cos β = b, show that
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.
Reduce each of the following expressions to the sine and cosine of a single expression:
cos x − sin 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 sin α − sin β = a and cos α + cos β = b, then write the value of cos (α + β).
If \[\tan A = \frac{a}{a + 1}\text{ and } \tan B = \frac{1}{2a + 1}\]
tan 3A − tan 2A − tan A =
If cot (α + β) = 0, sin (α + 2β) is equal to
If \[\tan\theta = \frac{1}{2}\] and \[\tan\phi = \frac{1}{3}\], then the value of \[\tan\phi = \frac{1}{3}\] is
If cos (A − B) \[= \frac{3}{5}\] and tan A tan B = 2, then
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 α and β are the solutions of the equation a tan θ + b sec θ = c, then show that tan (α + β) = `(2ac)/(a^2 - c^2)`.
If `(sin(x + y))/(sin(x - y)) = (a + b)/(a - b)`, then show that `tanx/tany = a/b` [Hint: Use Componendo and Dividendo].
Find the general solution of the equation `(sqrt(3) - 1) costheta + (sqrt(3) + 1) sin theta` = 2
[Hint: Put `sqrt(3) - 1` = r sinα, `sqrt(3) + 1` = r cosα which gives tanα = `tan(pi/4 - pi/6)` α = `pi/12`]
The value of `cot(pi/4 + theta)cot(pi/4 - theta)` is ______.
If tanα = `1/7`, tanβ = `1/3`, then cos2α is equal to ______.
State whether the statement is True or False? Also give justification.
If tanA = `(1 - cos B)/sinB`, then tan2A = tanB
