4.6 Trig Identities
Chapter
Chapter 4
Section
4.6
Lectures on Trig Identities 10 Videos

# Introduction to Trig Ratios and to Identity

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

7.22mins
Introduction to Trig Ratios and to Identity

## Simplifying \cot x \sin x

ex Simplify cot \theta \times \sin \theta.

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

= \cos \theta

0.33mins
Simplifying cotx sin x

## Simplifying Trig Expression 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

1.14mins
Simplifying Trig Expression 1+tan^2x

## Simplifying Trig Expression \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}

1.13mins
Simplifying Trig Expression 1+sinx:cos^2x

# Simplifying Trig Expression

\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}

1.05mins
Simplifying Trig Expression sinx +sin^2x:cosx1+sinx

## Introduction to Proving Identities

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}

2.36mins
Introduction to Proving Identities

## Proving Identity with Relative Angles

1.40mins
Proving Identity ex1

## Proving Identity ex2

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

1.17mins
Proving Identity ex2

## Proving Identity ex3

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}

0.55mins
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} 

1.20mins
Q1

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

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?

0.24mins
Q3a

Prove the identity:

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

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?

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?

0.25mins
Q3d

Prove the identity:

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

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?

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?

0.36mins
Q4c

Prove the identity:

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

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?

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?

1.00mins
Q6

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

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}.

1.20mins
Q8

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

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?

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?

1.01mins
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.

4.02mins
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.

4.02mins
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.

1.32mins
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?

1.40mins
Q17

If \sin^2\theta + \sin^22\theta+ \sin^23\theta = 1, what does \cos^23\theta + \cos^22\theta + \cos^2\theta equal?

1.02mins
Q21

Given \sin \theta\cot \theta = \frac{\sqrt{3}}{2}, find the value of \sin\theta .

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}