Write down the expansion of \sin \left(\alpha + \beta\right).
By replacing \beta with - \beta, determine the expansion of \sin \left(\alpha - \beta\right).
Rewrite the following as a function of x:
\sin \left(\dfrac{\pi}{2} - x\right)
\tan \left(\dfrac{3 \pi}{4} - x\right)
\cos \left(x - \dfrac{\pi}{4}\right)
Suppose \cos A = \dfrac{3}{5} and \sin B = \dfrac{15}{17} where 0 < A < \dfrac{\pi}{2} and 0 < B < \dfrac{\pi}{2}:
Find \sin A.
Find \cos B.
Find \sin \left(A - B\right).
Find \cos \left(A - B\right).
In which quadrant is angle \left(A - B\right)?
Find \tan \left(A - B\right).
Suppose \sin A = \dfrac{3}{5} and \sin B = - \dfrac{12}{13} where 0 < A < \dfrac{\pi}{2} and \pi < B < \dfrac{3 \pi}{2}:
Find \cos A.
Find \cos B.
Find \sin \left(A + B\right).
Find \sin \left(A - B\right).
Find \tan \left(A + B\right).
Find \tan \left(A - B\right).
In which quadrant is A + B?
In which quadrant is A - B?
Suppose \cos A = - \dfrac{3}{5} and \cos B = - \dfrac{8}{17} where \pi \lt A \lt \dfrac{3 \pi}{2} and \pi \lt B \lt \dfrac{3 \pi}{2}:
Find \sin A.
Find \sin B.
Find \sin \left(A + B\right).
Find \sin \left(A - B\right).
Find \tan \left(A + B\right).
Find \tan \left(A - B\right).
In which quadrant is A + B?
In which quadrant is A - B?
Given that \sin A = \dfrac{24}{25} and \tan B = \dfrac{20}{21} where A and B are acute angles:
Find the exact value of \tan \left(A - B\right).
Find the exact value of \tan \left(A + B\right).
For each of the following:
Write the angle as a sum or difference of two angles
Hence find the exact value of the ratio.
\cos \dfrac{7 \pi}{12}
\sin \dfrac{11 \pi}{12}
\tan \dfrac{\pi}{12}
\tan \left( - \dfrac{\pi}{12} \right)
Find the exact value of the following:
\sin \dfrac{5 \pi}{36} \cos \dfrac{\pi}{36} + \sin \dfrac{\pi}{36} \cos \dfrac{5 \pi}{36}
\cos \dfrac{21 \pi}{10} \cos \dfrac{\pi}{10} + \sin \dfrac{21 \pi}{10} \sin \dfrac{\pi}{10}
\dfrac{\tan \dfrac{2 \pi}{9} - \tan \dfrac{\pi}{18}}{1 + \tan \dfrac{2 \pi}{9} \tan \dfrac{\pi}{18}}
Express the following in simplest form:
\sin A \cos 2 B + \cos A \sin 2 B
\cos \left( 3 \theta + x\right) \cos 3 \theta - \sin \left( 3 \theta + x\right) \sin 3 \theta
\dfrac{\tan 4 m - \tan m}{1 + \tan 4 m \tan m}
\sin \left(\pi + \theta\right) - \sin \left(\pi - \theta\right)
Prove the following identities:
\sin \left(x + y\right) - \sin \left(x - y\right) = 2 \cos x \sin y
\dfrac{\sin \left(x - y\right)}{\cos x \cos y} = \tan x - \tan y
\dfrac{\cos \left(x + y\right)}{\cos y} - \dfrac{\sin \left(x + y\right)}{\sin y} = - \dfrac{\sin x}{\sin y \cos y}
\dfrac{\tan \left(x - y\right) + \tan y}{1 - \tan \left(x - y\right) \tan y} = \tan x
\tan \left(\theta + \alpha\right) \tan \left(\theta - \alpha\right) = \dfrac{\tan ^{2}\left(\theta\right) - \tan ^{2}\left(\alpha\right)}{1 - \tan ^{2}\left(\theta\right) \tan ^{2}\left(\alpha\right)}
\dfrac{\tan A - \tan B}{\tan A + \tan B} = \dfrac{\sin \left(A - B\right)}{\sin \left(A + B\right)}
Consider 2 x = x+x, and hence verify the following:
\cos 2 x = 2 \cos ^{2}\left(x\right) - 1
\sin 2 x = 2 \sin x \cos x
If \tan A + \tan B = - 5 and \tan A \tan B = 6, prove that \sin \left(A + B\right) = \cos \left(A + B\right).
The pendulum in a grandfather clock is pulled to its maximum displacement on one side and then released. The displacement, in centimetres, of the pendulum from its vertical equilibrium position is given by the formula: D = 17 \sin \left( \pi \left(x + \dfrac{1}{2}\right)\right)where x is the time in seconds since the pendulum was released.
Find the maximum displacement of the pendulum.
Rewrite the formula for D in terms of the cosine function.
Find the period of oscillation of the pendulum. That is, how long does the pendulum take to return to the maximum displacement from where it started?
The average depth of water, in metres, of a certain harbour can be modelled by the function: h = \dfrac{1}{2} \sin \left( \dfrac{\pi}{6} x + \dfrac{2 \pi}{3}\right) + \dfrac{5}{2}where x is the time in hours since midnight. The average depth of water has been graphed over a 24-hour period.
Using the graph, determine the best estimate of the average depth of the water at 3 am, correct to one decimal place.
Calculate the average depth of the water in the harbour at 5:00 pm.
Rewrite the function in expanded form, in terms of \sin \left( \dfrac{\pi}{6} x\right) and \cos \left( \dfrac{\pi}{6} x\right).