6.05 Double angle identities
Evaluate:
Use the double angle identity for the cosine ratio to simplify the following expression:
10 \cos ^{2}\left(\dfrac{5 \pi}{8}\right) - 5Given \cos \theta = \dfrac{4}{5} and \sin \theta <0, find:
\sin \theta
\sin 2 \theta
\cos 2 \theta
Given \sin \theta = - \dfrac{7}{25} and \cos \theta <0, find:
\cos \theta
\sin 2 \theta
\cos 2 \theta
Given \cos \theta = \dfrac{4}{5} and \sin \theta = \dfrac{3}{5}, find:
\sin 2 \theta
\cos 2 \theta
Consider the expression \dfrac{1 + \cos x}{2} where \sin x = - \dfrac{12}{13} and \dfrac{3\pi}{2} < x < 2\pi.
Simplify \dfrac{1 + \cos x}{2}.
Find \cos x.
Hence find \cos \left(\dfrac{x}{2}\right).
Consider the expression \dfrac{1 + \cos x}{2} where \sin x = \dfrac{5}{13} and \dfrac{\pi}{2} < x < \pi.
Simplify \dfrac{1 + \cos x}{2}.
Find \cos x.
Hence find \cos \left(\dfrac{x}{2}\right).
Consider the expression \dfrac{1 - \cos x}{2} where \tan x = 4 and 0 < x < \dfrac{\pi}{2}.
Simplify \dfrac{1 - \cos x}{2}.
Find \cos x.
Hence find \sin \left(\dfrac{x}{2}\right).
Find the exact value of the following:
Find the exact value of the following with the given conditions:
\sin \left(\dfrac{x}{2}\right) if \cos x = - \dfrac{4}{7} and \dfrac{\pi}{2} < x < \pi
\cos \left(\dfrac{x}{2}\right) if \cos x = \dfrac{2}{5} and 0 < x <\dfrac{\pi}{2}
\cos x if \cos 2 x = \dfrac{2}{3} and \pi < x <\dfrac{3 \pi}{2}
\sin x if \cos 2 x = - \dfrac{4}{7} and \dfrac{3 \pi}{2} < x <2 \pi
If \tan x = \dfrac{p}{q}, express p \sin 2 x + q \cos 2 x in terms of p and q only.
Consider the expression \cos 3 \theta.
By writing \cos 3 \theta as \cos \left( 2 \theta + \theta\right), show that \cos 3 \theta = 4 \cos ^{3}\left(\theta\right) - 3 \cos \theta.
Using the fact that \sin 36 \degree = \cos 54 \degree, solve for the exact value of \sin 18 \degree.
Prove the following by simplifying the left hand side (\text{LHS}) of the identity:
Solve the following equations. Round your answer to one decimal place when necessary.
\cos 2 x = 3 \cos x - 2 for 0 \leq x < 2 \pi
\sin 2 x - \cos x = 0 for 0 \leq x \leq 2\pi
3 \sin x - 1 = 1 - \cos 2 x for 0 \leq x \leq 2\pi
\cos ^{2}\left(x\right) = 2 \cos ^{2}\left(\dfrac{x}{2}\right) for 0 \degree \leq x \leq 360 \degree
8 \sin ^{2}\left(x\right) - 4 \sin 2 x = 8 for 0 \degree \leq x \leq 360 \degree
Consider the identity \sin \left(\theta + 2 \theta\right) = \sin \theta \cos 2 \theta + \cos \theta \sin 2 \theta.
Prove that \sin 3 \theta = 3 \sin \theta - 4 \sin ^{3}\left(\theta\right).
Hence solve the equation \sin 3 \theta = 2 \sin \theta for 0 \leq \theta \leq \pi.
Consider the equation x^{2} - \sqrt{2} \sin \left(45 \degree + \theta\right) x + \sin \theta \cos \theta = 0.
Find the value of the discriminant of the equation.
What can be said about the equation's solutions?
The speed of an aircraft can be measured by a 'Mach number', m. The Mach number is the ratio of the speed of an aircraft to the speed of sound. For example, a speed of Mach 2 means that the aircraft is travelling at 2 times the speed of sound.
If an aircraft is travelling faster than the speed of sound (m > 1), it projects sound waves that form a cone behind it. If the angle at the vertex of the cone is \theta, then
\sin \left(\dfrac{\theta}{2}\right) = \dfrac{1}{m}
If m = \dfrac{3}{2}, then find the value of \theta to the nearest degree.
If \theta = 90, then find the value of m to one decimal place.
The power, P, consumed by an electrical device can be calculated by taking the product of the current, I, and voltage, V, in its circuit. That is, P = V I.
When a light is turned on, a household circuit has current I = \dfrac{5}{4} \sin 110 \pi t and voltage V = 151 \sin 110 \pi t at time t.
Find the equation for the power consumed by the light, P, in terms of t.
Use your CAS calculator to graph the function P(t), and find the maximum and minimum values for power consumed by the light.
Using a double angle rule, express P in the form P = a \cos b \pi \theta + c.
The power rating of a lightbulb is the average of the maximum and minimum amount of power that is used by the lightbulb, rounded to the nearest 10 watts. Find the power rating of the lightbulb.
A rectangle is inscribed in a semicircle of radius 1, as shown in the diagram:
Express x in terms of a.
Express y in terms of a.
Let A be the area of the triangle. Show that A = \sin 2 a.
Notice that a must be between 0 \degree and 90 \degree. What value of a makes the area of the rectangle largest?
Hence, find the value for y which results in the largest possible area. Give your answer as an exact value with a rational denominator.
Hence, find the length of the rectangle which results in the largest possible area. Give your answer as an exact value with a rational denominator.