7.3 Double Angle Formulas
Chapter
Chapter 7
Section
7.3
2. Half Angles 3 Videos

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}

3.33mins
Introduction to Half Angles

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

3.12mins
Finding sin(x/2) and cos(x/2) given cos x

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

2.38mins
Finding tan(x/2) given sin x
1. Double Angles 9 Videos

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} 

1.37mins
Introduction to Double Angle Formulas Sine Double Angle Formula

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} 

1.57mins
Cosine Double Angle formulas

Prove that \frac{\sin 3x}{\sin x \cos x} = 4 \cos x - \sec x.

3.41mins
Trig Identity using Double Angles

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} 

4.27mins
Lowering Power using Double Angle formulas
Solutions 46 Videos

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 

0.56mins
Q1ab

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} 

0.24mins
Q1cd

Express it as a single trigonometric ratio.

 \displaystyle 4 \sin x \cos x 

0.17mins
Q1e

Express it as a single trigonometric ratio.

 \displaystyle 2 \cos^2\frac{\theta}{2} - 1 

0.19mins
Q1f

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle 2\sin 45^o\cos 45^o 

0.21mins
Q2a

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle \cos^230^o -\sin^230^o 

0.46mins
Q2b

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle 2\sin\frac{\pi}{12} \cos\frac{\pi}{12} 

0.15mins
Q2c

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle \cos^2\frac{\pi}{12} - \sin^2\frac{\pi}{12} 

0.24mins
Q2d

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle 1 - 2\sin^2\frac{3\pi}{8} 

0.11mins
Q2e

Express it as a single trigonometric ratio and then evaluate.

 \displaystyle 2 \tan 60^o \cos^2 60^o 

1.28mins
Q2f

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \sin 4\theta 

1.12mins
Q3a

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \cos 3x 

1.02mins
Q3b

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \tan x 

0.36mins
Q3c

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \cos 6\theta 

0.56mins
Q3d

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \sin x 

0.34mins
Q3e

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \tan 5\theta 

0.50mins
Q3f

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}

4.39mins
Q4

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

3.03mins
Q5

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

4.22mins
Q6

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

1.54mins
Q7

Determine the values of a in the following equation: 2\tan x - \tan 2x + 2a = 1 - \tan 2x \tan^2x.

4.29mins
Q8

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?

1.54mins
Q9

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?

5.27mins
Q10

Use a double angle formula to develop a formula for \sin 4x in terms of x.

0.47mins
Q11a

Verify that  \displaystyle \sin \frac{2\pi}{3} = \sin \frac{8\pi}{3} 

1.54mins
Q11b

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

2.42mins
Q12a

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

2.12mins
Q12b

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?

3.47mins
Q12c

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 

2.00mins
Q13a

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 

0.35mins
Q13b

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} 

1.50mins
Q13c

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 

1.29mins
Q13d

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].

4.32mins
Q14

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

0.41mins
Q15a

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

0.45mins
Q15b

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}

0.26mins
Q15c

Eliminate A from each pair of equations to find an equation that relates x to y.

x =\tan 2A, y = \tan A

0.30mins
Q16a

Eliminate A from each pair of equations to find an equation that relates x to y.

x =\cos 2A, y = \cos A

1.16mins
Q16b

Eliminate A from each pair of equations to find an equation that relates x to y.

x =\cos 2A, y = \csc A

0.44mins
Q16c

Eliminate A from each pair of equations to find an equation that relates x to y.

x =\sin 2A, y = \sec 4A

1.48mins
Q16d

Solve each equation for values of x in the interval 0\leq x \leq 2\pi

\cos 2x = \sin x

1.39mins
Q17a

Solve each equation for values of x in the interval 0\leq x \leq 2\pi

\sin 2x - 1 =\cos 2x

3.26mins
Q17b

Convert the expression into ratio of tangent.

\sin 2\theta

1.42mins
Q18a

Convert the expression into ratio of tangent.

\cos 2\theta

2.07mins
Q18b

Convert the expression into ratio of tangent.

\frac{\sin 2\theta}{1 + \cos 2\theta}

\frac{1 - \cos 2\theta}{\sin 2\theta}