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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

Cos3x in terms of Cos x only

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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}
```

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Lowering Power using Double Angle formulas

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Simplifying Expressions using Double Angles

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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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}
```

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Q1cd

Express it as a single trigonometric ratio.

```
\displaystyle
4 \sin x \cos x
```

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0.17mins

Q1e

Express it as a single trigonometric ratio.

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

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0.19mins

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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0.15mins

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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Q2d

Express it as a single trigonometric ratio and then evaluate.

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

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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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1.02mins

Q3b

Use a double angle formula to rewrite the trigonometric ratio.

```
\displaystyle
\tan x
```

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0.36mins

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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0.34mins

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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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`

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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`

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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`

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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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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?

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5.27mins

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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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`

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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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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?

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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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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
```

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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}
```

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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
```

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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]`

.

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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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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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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}`

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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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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`

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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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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`

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1.48mins

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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1.39mins

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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3.26mins

Q17b

Convert the expression into ratio of tangent.

`\sin 2\theta`

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1.42mins

Q18a

Convert the expression into ratio of tangent.

`\cos 2\theta`

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2.07mins

Q18b

Convert the expression into ratio of tangent.

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

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0.37mins

Q18c

Convert the expression into ratio of tangent.

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

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Q18d