7.3 Double Angle Formulas
Chapter
Chapter 7
Section
7.3
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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}

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Introduction to Half Angles

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

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

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

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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} 

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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} 

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Cosine Double Angle formulas

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

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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} 

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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 

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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} 

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Q1cd

Express it as a single trigonometric ratio.

 \displaystyle 4 \sin x \cos x 

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Q1e

Express it as a single trigonometric ratio.

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

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Q1f

Express it as a single trigonometric ratio and then evaluate.

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

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Q2a

Express it as a single trigonometric ratio and then evaluate.

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

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Q2b

Express it as a single trigonometric ratio and then evaluate.

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

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Q2c

Express it as a single trigonometric ratio and then evaluate.

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

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0.24mins
Q2d

Express it as a single trigonometric ratio and then evaluate.

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

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0.11mins
Q2e

Express it as a single trigonometric ratio and then evaluate.

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

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1.28mins
Q2f

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \sin 4\theta 

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1.12mins
Q3a

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \cos 3x 

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Q3b

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \tan x 

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Q3c

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \cos 6\theta 

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Q3d

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \sin x 

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Q3e

Use a double angle formula to rewrite the trigonometric ratio.

 \displaystyle \tan 5\theta 

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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}

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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

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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

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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

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Q7

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

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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?

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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?

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Q10

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

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Q11a

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

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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

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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

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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?

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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 

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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 

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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} 

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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 

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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].

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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

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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

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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}

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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

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Q16a

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

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

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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

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Q16c

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

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

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Q16d

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

\cos 2x = \sin x

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Q17a

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

\sin 2x - 1 =\cos 2x

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Q17b

Convert the expression into ratio of tangent.

\sin 2\theta

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Q18a

Convert the expression into ratio of tangent.

\cos 2\theta

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Q18b

Convert the expression into ratio of tangent.

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

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Q18c

Convert the expression into ratio of tangent.

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

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Q18d