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Lectures on Trig Identities
10 Videos

Trig Ratios

`\sin \theta = \dfrac{opp}{hyp}`

`\cos \theta = \dfrac{adj}{hyp}`

`\tan \theta = \dfrac{opp}{adj}`

Reciprocal Identities

`\csc \theta = \dfrac{1}{\sin \theta}`

`\sec \theta = \dfrac{1}{\cos \theta}`

`\cot \theta = \dfrac{1}{\tan \theta}`

Pythagorean Identities

`\sin^2 \theta + \cos^2 \theta = 1`

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7.22mins

Introduction to Trig Ratios and to Identity

`\cot x \sin x`

*ex* Simplify `cot \theta \times \sin \theta`

.

`= \dfrac{\cos \theta}{\sin \theta} \times \sin \theta`

`= \cos \theta`

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0.33mins

Simplifying cotx sin x

`1+\tan^2x`

*ex* `1+\tan^2x`

`= 1+ (\dfrac{\sin \theta}{\cos \theta})^2`

`= 1 + \dfrac{\sin^2 \theta}{\cos^2 \theta}`

`= \dfrac{\cos^2 \theta + \sin^2 \theta}{\cos^2 \theta}`

`= \dfrac{1}{\cos^2 \theta}`

`= \sec^2 \theta`

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1.14mins

Simplifying Trig Expression 1+tan^2x

`\dfrac{1 + \sin \theta}{\cos^2 \theta}`

*ex* Simplify `\dfrac{1 + \sin \theta}{\cos^2 \theta}`

`= \dfrac{1 + \sin \theta}{1 - \sin^2 \theta}`

`= \dfrac{1 + \sin \theta}{(1 + \sin \theta)(1 - \sin \theta)}`

`= \dfrac{1}{1 - \sin \theta}`

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Simplifying Trig Expression 1+sinx:cos^2x

`\displaystyle \begin{aligned} & \tan ^{2} \theta-\sec ^{2} \theta \=& \frac{\sin ^{2} \theta}{\cos ^{2} \theta}-\frac{1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-1}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\left(\sin ^{2} \theta+\cos ^{2} \theta\right)}{\cos ^{2} \theta} \=& \frac{\sin ^{2} \theta-\sin ^{2} \theta-\cos ^{2} \theta}{\cos ^{2} \theta}=-1 \end{aligned} `

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Simplifying Trig Expression sinx +sin^2x:cosx1+sinx

LS = RS

Prove that LS is the same as RS.

*ex* `\dfrac{1}{\cos \theta} + \tan \theta = \dfrac{1 + \sin \theta}{\cos \theta}`

`\displaystyle \begin{aligned} L S &=\frac{1}{\cos \alpha}+\frac{\sin \alpha}{\cos \alpha} \\ &=\frac{1+\sin \alpha}{\cos \alpha}=R .5 . \end{aligned} `

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Introduction to Proving Identities

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Proving Identity ex1

*ex* Prove `2 \cos^2 \theta + \sin^2 \theta - 1 = \cos^2 \theta`

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Proving Identity ex2

*ex* Prove `\tan \theta + \dfrac{1}{\tan \theta} = \dfrac{1}{\sin \theta \cos \theta}`

`\displaystyle \begin{aligned} & \tan \theta+\frac{1}{\tan \theta}=\frac{1}{\sin \theta \cos \theta} \\ L S &=\frac{\sin \theta}{\cos \theta} s+\frac{\cos \theta}{\sin \theta} \frac{c}{c} \\ &=\frac{\sin ^{2} \theta+\cos ^{2} \theta}{\sin \theta \cos \theta} \\ &=\frac{1}{\sin \theta \cos \theta}=R S \end{aligned} `

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Proving Identity ex3

Solutions
24 Videos

For the proof of `\sin^2B + \cos^2B = 1`

using the triangle:

Fill in the missing step to prove.

```
\displaystyle
\begin{array}{lllll}
& LS &= (\frac{y}{r})^2 + (\frac{x}{r})^2\\
& &= \frac{y^2}{r^2} + \frac{x^2}{r^2}\\
& &= \frac{y^2+ x^2}{r^2} \\
& &= ......... \\
& &=1
\end{array}
```

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1.20mins

Q1

Graph the relation `y = \sin^2x + \cos^2x`

.

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0.20mins

Q2

Prove the identity:

`\sin \theta = \cos \theta \tan \theta `

Which of the following could be a step going from right to left?

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0.24mins

Q3a

Prove the identity:

```
\displaystyle
\csc \theta = \sec \theta \cot \theta
```

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0.22mins

Q3b

Prove the identity:

```
\displaystyle
\cos \theta = \sin \theta \cot \theta
```

Which of the following could be a step going from right side to left?

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0.15mins

Q3c

Prove the identity:

```
\displaystyle
\sec \theta = \csc \theta \tan \theta
```

Which of the following could be a step going from right side to left?

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0.25mins

Q3d

Prove the identity:

```
\displaystyle
1 +\csc \theta = \csc \theta(1 + \sin \theta)
```

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0.46mins

Q4a

Prove the identity:

```
\displaystyle
\cot \theta \sin \theta \sec \theta = 1
```

Which of the following could be a step going from left side to right?

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0.22mins

Q4b

Prove the identity:

```
\displaystyle
\cos \theta(\sec \theta - 1) = 1 -\cos \theta
```

Which of the following could be a step going from left side to right?

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0.36mins

Q4c

Prove the identity:

```
\displaystyle
1 + \sin \theta = \sin \theta(1 + \csc \theta)
```

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0.38mins

Q4d

Prove that ```
\displaystyle
1 - \sin^2\theta = \sin \theta \cos \theta \cot \theta
```

Which of the following could be a step going from right side to left?

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0.48mins

Q5

Prove that ```
\displaystyle
\csc^2\theta = \cot^2\theta + 1
```

Which of the following could be a step going from left side to right?

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Q6

Prove that `\displaystyle \frac{\cos \theta}{1 + \sin \theta} = \frac{1 - \sin \theta}{\cos \theta}`

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1.03mins

Q7

Start the proof by adding the rational expression on the left for `\displaystyle \frac{\cos \theta}{1 - \sin \theta} + \frac{\cos \theta}{1 + \sin \theta}= \frac{2}{\cos \theta}`

.

Show your step and simply.

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Q8

Prove that `\displaystyle \csc^2\theta \cos^2\theta = \csc^2\theta - 1`

.

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0.57mins

Q9

Prove that `\displaystyle \tan \theta +\cot \theta = \frac{\sec \theta}{\sin \theta}`

.

Which of the following could be a step going from left side to right?

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0.56mins

Q12

Prove that `\displaystyle \cot^2 \theta(1 +\tan^2\theta) =\csc^2\theta`

.

Which of the following could be a step going from left side to right?

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Q13

You are on the last leg of the orienteering course. Determine the direction and distance from the information given. Draw the leg on your map. Label all angles and distances.

Direction: East of south

Determine the two angles between 0° and `360^o`

such that `\frac{\csc \theta}{\sec \theta} = \cot \theta`

. Add their degree measures and divide by 9. Use this angle.

*Hint* Use identities to simplify each side of the equation first.

Distance: The result of evaluating

```
\displaystyle
20(\sec^2\theta \sin^2\theta + \sec^2\theta \cos^2\theta -\tan^2\theta\sin^2\theta - \tan^2\theta \cos^2\theta)
```

, founded to the nearest metre if necessary.

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Q14

Draw a right angle. Using the vertex as the centre, draw a quarter-circle that intersects the two arms of the angle. Select any point `A`

on the quarter-circle, other than a point on one of the arms. Draw a tangent line to the quarter-circle at `A`

such that the line intersects one arm at point B and the other arm at point C. Label `\angle AOC`

as 6. Show that the measure of BA divided by the radius of the quarter-circle equals the cotangent of `\angle \theta`

.

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Q15

A student writes the following proof for the identity `\cos \theta = \sin \theta \cot \theta`

. Critique it for form, and rewrite it in proper form.

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Q16

Consider the equation
`\displaystyle \tan^2\theta -\sin^2\theta = \sin^2\theta\tan^2\theta`

.

**a)** Which of the basic identities will you use first to simplify the left side?

**b)** Which is the one step before simplifying to the right side?

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Q17

If `\sin^2\theta + \sin^22\theta+ \sin^23\theta = 1`

, what does `\cos^23\theta + \cos^22\theta + \cos^2\theta`

equal?

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Q21

Given `\sin \theta\cot \theta = \frac{\sqrt{3}}{2}`

, find the value of `\sin\theta`

.

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Q22

Given `\cos \theta = 3\sin \theta`

, a possible value for `\cos^2\theta`

is

```
\displaystyle
\begin{array}{cccccc}
&(A)&0.25 &(B)& 0.7071 &(C) & 0.5 &(D) & 0.9 &(E) & 1 \\
\end{array}
```

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Q23