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}

Summary of Half Angle Formulas

\displaystyle \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}

\displaystyle \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}

\displaystyle \tan \frac{x}{2} = \frac{\sin x }{1 + \cos x}

Finding \sin(x/2) and \cos(x/2) given \cos x

Finding \tan(x/2) given \sin x

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\sin\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}

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}

Simplifying using Cosine Double Angle Fomula examples

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}

Simplifying Expressions using Double Angles

Trig Identity using Tangent Double Angle Example