Introduction to Half Angles
\displaystyle
\begin{array}{lllll}
\sin \frac{x}{2} &= \pm \sqrt{\frac{1 - \cos x}{2}}\\
\cos \frac{x}{2} &= \pm \sqrt{\frac{1 + \cos x}{2}}\\
\tan \frac{x}{2} &= \frac{1- \cos x}{\sin x}
\end{array}
Whether it's + or - depends on the size of angle \frac{x}{2}
Finding \sin(x/2) and \cos(x/2) given \cos x
Finding \tan(x/2) given \sin x
Introduction to Double Angle Formulas Sine Double Angle Formula
\displaystyle
\begin{array}{llll}
\sin 2x &= 2 \sin x \cos x \text{Formula for sine} \\
\cos 2x &= \cos^2 x - \sin^2 x \text {Formula for cosine} \\
&= 1 - 2\sin^2 x \\
&= 2\cos^2 x - 1 \\
\tan 2x &= \frac{2\tan x}{1 - \tan^2x} \text{Formula for tangent}
\end{array}
Cosine Double Angle formulas
\displaystyle
\begin{array}{lllll}
\cos 2x &= \cos^2 x - \sin^2 x \\
&= 1 - 2\sin^2 x \\
&= 2\cos^2 x - 1 \\
\end{array}
Prove that \frac{\sin 3x}{\sin x \cos x} = 4 \cos x - \sec x.
Cos3x in terms of Cos x only
Lowering Power using Double Angle formulas
\displaystyle
\begin{array}{lllll}
\sin^2 x &= \frac{1 - \cos 2x}{2} \\
\cos^2 x &= \frac{1 + \cos 2x}{2} \\
\tan^2x &= \frac{1 - \cos 2x}{1 + \cos 2x}
\end{array}
Single to Double Angle
Express each of the following as a single trigonometric ratio.
(a)
\displaystyle
2\sin 5x \cos 5x
(b)
\displaystyle
\cos^2\theta -\sin^2\theta
Express each of the following as a single trigonometric ratio.
a)
\displaystyle
1 -2\sin^23x
b)
\displaystyle
\frac{2\tan 4x}{1- \tan^24x}
Express it as a single trigonometric ratio.
\displaystyle
4 \sin x \cos x
Express it as a single trigonometric ratio.
\displaystyle
2 \cos^2\frac{\theta}{2} - 1
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
2\sin 45^o\cos 45^o
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
\cos^230^o -\sin^230^o
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
2\sin\frac{\pi}{12} \cos\frac{\pi}{12}
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
\cos^2\frac{\pi}{12} - \sin^2\frac{\pi}{12}
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
1 - 2\sin^2\frac{3\pi}{8}
Express it as a single trigonometric ratio and then evaluate.
\displaystyle
2 \tan 60^o \cos^2 60^o
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\sin 4\theta
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\cos 3x
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\tan x
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\cos 6\theta
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\sin x
Use a double angle formula to rewrite the trigonometric ratio.
\displaystyle
\tan 5\theta
Determine the values of \sin 2\theta, \cos 2\theta and \tan 2\theta, given \cos \theta = \frac{3}{5} and 0 \leq \theta \leq \frac{\pi}{2}
Determine the values of \sin 2\theta, \cos 2\theta and \tan 2\theta, given \tan \theta = -\frac{7}{24} and \frac{3\pi}{2} \leq \theta \leq 2\pi
Determine the values of \sin 2\theta, \cos 2\theta and \tan 2\theta, given \sin \theta = -\frac{12}{13} and \frac{3\pi}{2} \leq \theta \leq 2\pi
Determine the values of \sin 2\theta, \cos 2\theta and \tan 2\theta, given \cos\theta = -\frac{4}{5} and \frac{\pi}{2} \leq \theta \leq \pi
Determine the values of a in the following equation: 2\tan x - \tan 2x + 2a = 1 - \tan 2x \tan^2x.
Jim needs to find the sine of \frac{\pi}{8}. If he knows that \cos \frac{\pi}{4} =\frac{1}{\sqrt{2}}, how can he use this fact to find the sine of \frac{\pi}{8}? What is his answer?
Maria needs to find the cosine of \frac{\pi}{12}. If she knows that \frac{\pi}{6} = \frac{\sqrt{3}}{2}, how can she use this fact to find the cosine of \frac{\pi}{12}? What is her answer?
Use a double angle formula to develop a formula for \sin 4x in terms of x.
Verify that
\displaystyle \sin \frac{2\pi}{3} = \sin \frac{8\pi}{3}
Use the appropriate compound angle formula and double angle formula to develop a formula for
\sin 3x in terms of \cos x and \sin x
Use the appropriate compound angle formula and double angle formula to develop a formula for
\cos 3x in terms of \cos x and \sin x
Use the appropriate compound angle formula and double angle formula to develop a formula for
\tan 3x in terms of \tan x
Which formula is correct?
The angel x lies in there interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{8}{9}. Without using a calculator, determine the value of
\displaystyle
\sin 2x
The angle x lies in there interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{8}{9}. Without using a calculator, determine the value of
\displaystyle
\cos 2x
The angel x lies in there interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{8}{9}. Without using a calculator, determine the value of
\displaystyle
\cos \frac{x}{2}
The angle x lies in there interval \frac{\pi}{2} \leq x \leq \pi, and \sin^2x = \frac{8}{9}. Without using a calculator, determine the value of
\displaystyle
\sin 3x
Create a flow chart to show how you would evaluate \sin 2a, given the value of \sin a, if a \in [\frac{\pi}{2}, \pi].
Describe how you could use your knowledge of double angle formulas to sketch the graph of each function. Include a sketch with your description.
f(x) =\sin x \cos x
Describe how you could use your knowledge of double angle formulas to sketch the graph of each function. Include a sketch with your description.
f(x) = 2\cos^2 x
Describe how you could use your knowledge of double angle formulas to sketch the graph of each function. Include a sketch with your description.
f(x) = \frac{\tan x}{1 -\tan^2x}
Eliminate A from each pair of equations to find an equation that relates x to y.
x =\tan 2A, y = \tan A
Eliminate A from each pair of equations to find an equation that relates x to y.
x =\cos 2A, y = \cos A
Eliminate A from each pair of equations to find an equation that relates x to y.
x =\cos 2A, y = \csc A
Eliminate A from each pair of equations to find an equation that relates x to y.
x =\sin 2A, y = \sec 4A
Solve each equation for values of x in the interval 0\leq x \leq 2\pi
\cos 2x = \sin x
Solve each equation for values of x in the interval 0\leq x \leq 2\pi
\sin 2x - 1 =\cos 2x
Convert the expression into ratio of tangent.
\sin 2\theta
Convert the expression into ratio of tangent.
\cos 2\theta
Convert the expression into ratio of tangent.
\frac{\sin 2\theta}{1 + \cos 2\theta}
Convert the expression into ratio of tangent.
\frac{1 - \cos 2\theta}{\sin 2\theta}