Basic Trig Identities

Basic Trig Identities

Reciprocal, Quotient, Pythagorean

What are Trig Identities?

• Trig identities are expressions that are true for ALL angles

• Ex.  $\sin x=\frac{1}{2}$  is an equation that is only true for some angles, but  $\sin x=\frac{1}{\csc x}$  is an identitiy because it is true for all angles of x.

Identity Categories:

• Reciprocal

• Quotient

• Pythagorean

• Sum & Difference

• Double Angle

• Half Angle

• For this part of our unit, we will be discussing Reciprocal, Quotient and Pythagorean Identities **MEMORIZE THEM!

Reciprocal Identities: 

•  $\sin x=\frac{1}{\csc x}\ \ \ \ \ \cos x\ =\ \frac{1}{\sec x}\ \ \ \ \tan x\ =\ \frac{1}{\cot x}$  

•  $\csc x=\frac{1}{\sin x}\ \ \ \ \ \sec x\ =\ \frac{1}{\cos x}\ \ \ \ \ \cot x=\frac{1}{\tan x}$  

• These would also be true if all trig functions were raised to the same power!                        Ex:   $\sin^2x=\frac{1}{\csc^2x}$  

Quotient Identities:

 $\tan x=\frac{\sin x}{\cos x}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \cot x=\frac{\cos x}{\sin x}$  

Again, these would also be true if they were all raised to the same power.  
Ex:       $\tan^2x=\frac{\sin^2x}{\cos^2x}$  

Multiple Select

Which statements are true?

• $\sin x=\frac{1}{\cos x}$

• $\tan^2x=\frac{\sin^2x}{\cos x}$

• $\sec x=\frac{1}{\cos x}$

• $\cot x=\frac{\cos x}{\sin x}$

• $\csc^2x=\frac{1}{\sin^2x}$

Pythagorean identity #1:

• Think about the pythagorean equation for a unit circle where r = 1. =======>

• This would be:        $\sin^2x+\cos^2x=1$ 

• This first identity is probably the most used!   Memorize for life!!

• The other 2 pythagorean identities are derived from this expression

Pythagorean Identity #2

• We can derive another identity by dividing each term in the previous expression by (sin^2x):

•  $\frac{\sin^2x}{\sin^2x}+\frac{\cos^2x}{\sin^2x}=\frac{1}{\sin^2x}$  

• Simplify to get=====>           $1+\cot^2x=\csc^2x$  

Pythagorean Identity #3

• Another identity can be derived by dividing each term in the original expression by (cos^2x):

•  $\frac{\sin^2x}{\cos^2x}+\frac{\cos^2x}{\cos^2x}=\frac{1}{\cos^2x}$  

• Simplify to get=====>           $\tan^2x+1=\sec^2x$  

Anymore Pythagorean Identities??

Yes! Even more pythagorean identities can be obtained by rearranging (adding/subtracting) within the 3 identities we just covered!

See if you can figure out what they are from the following questions....

Multiple Choice

Rearrange:  $\sin^2x+\cos^2x=1$  to solve for:   $\sin^2x$   

•  $\sin^2x=1-\cos^2x$  

•  $\sin^2x=1+\cos^2x$  

•  $\sin^2x=\cos^2x-1$  

Multiple Choice

Rearrange:  $\sin^2x+\cos^2x=1$  to solve for  $\cos^2x$  

•  $\cos^2x=1-\sin^2x$  

•  $\cos^2x=1+\sin^2x$  

•  $\cos^2x=\sin^2x-1$  

Multiple Choice

Rearrange to solve for $\tan^2\theta$ : 
                                        $1\ +\ \tan^2\theta=\ \sec^2\theta$  

• tan²θ = sec²θ + 1

• tan²θ = sec²θ - 1

• tanθ = secθ - 1

• tan²θ = opp/adj

Multiple Choice

Rearrange so it equals 1: 
                                        $1\ +\ \tan^2x=\ \sec^2x$  

•  $\tan^2x-\sec^2x=1$  

•  $\sec^2x-\tan^2x=1$  

•  $\sec^2x+\tan^2x=1$  

•  $\tan^2x-1=\sec^2x$  
  

Multiple Choice

Solve the following for  $\cot^2\theta$  :
 $1+\cot^2\theta=\csc^2\theta$  

• 1 = cot²θ + csc²θ

• cot²θ =1 - csc²θ

• cot²θ = csc²θ -1

• cot²θ = adj/opp

Multiple Choice

Rearrange so it equals 1 :
 $1+\cot^2\theta=\csc^2\theta$  

•  $1=\cot^2\theta+\csc^2\theta$ 

•  $\frac{\cot^2\theta}{\csc^2\theta}=1$  

•  $\cot^2\theta-\csc^2\theta=1$  
  

•  $\csc^2\theta-\cot^2\theta=1$  

IN CASE YOU DIDN'T GET THEM....HERE ARE ALL THE PYTHAGOREAN ARRANGEMENTS!

How do we use these identities?

• We can simplify trig expressions by substituting identities and using algebra

• Look at the example ======> 

• cscx and tanx were replaced with identities that helped reduce the expression down to one term:  $\sec x$  

• Let's try some problems!

Multiple Choice

Exchange each term with an identity then use algebra to reduce:

                                 $\frac{\tan x}{\sec x}$ 

•  $\cos x$  

•  $\csc x$  

•  $\cot x$  

•  $\sin x\$  

Solution Recap for:  $\frac{\tan x}{\sec x}=\sin x$  

•   Substitute with identities:   $\frac{\tan x}{\sec x}=\frac{\frac{\sin x}{\cos x}}{\frac{1}{\cos x}}$  

• Flip the bottom fraction and multiply.  Then reduce: $\frac{\sin x}{\cos x}\cdot\frac{\cos x}{1}=\sin x$  

Multiple Choice

Now try this one.  Again reduce by substituting with identities:
 $\left(\sec x\right)\left(\cot x\right)\left(\sin x\right)$ 

• sin x

• - sin x

• 1

• -1

Simplifying Suggestions

• Here are some common techniques for simplifying trig expressions

• Notice you will be using algebra to help reduce

• Let's look at more problems using these suggestions......

Multiple Choice

Reduce by substituting with identities:

 $\tan\ x\ \cot x\ -\cos^2x$  

• tanx

• cotx

• sin²x

• cos²x

Solution Recap for:

 $\tan x\cot x-\cos^2x=\ \sin^2x$ 

Replace with sinx & cosx identities:

 $\left(\frac{\sin x}{\cos x}\right)\left(\frac{\cos x}{\sin x}\right)-\left(\cos^2x\right)$  
Cross-divide out the cosx to get:   $1-\cos^2x$  which is the identity for:   $\sin^2x$  

Now try some on your own using the hints provided....

Multiple Choice

Simplify by factoring out a GCF first. Look at what you have left.  Can you replace with an identity?

 $\cos^3x+\sin^2x\cos x$ 

•  $\sin x$  

•  $\cos x$  

•  $\tan x$  

• 1

Multiple Choice

Distribute then exchange with identities:
 $\sin x\left(\sec x-\csc x\right)$ 

•  $1-\tan x$  

•  $\tan x-1$  

•  $1-\cot x$  

•  $\sin x$  

Multiple Choice

Simplify by first splitting into 2 fractions: (Note:  you CANNOT cross out anything until you've done this!)

 $\frac{\cos x-\sin x}{\sin x\cos x}$  

•  $\csc x-\sec x$  

• 1

• 0

•  $\sec x-\csc x$  

Multiple Choice

Substitute then find a common denominator so you can combine into 1 fraction:              $\cos x+\sin x\tan x$  

• csc x

• sec x

• 1/sec x

• cos x

Multiple Choice

Exchange each with identities, then get a common denom. on the bottom.  Once you have a complex fraction, flip and multiply.

    $\frac{\csc x\cos x}{\tan x+\cot x}$  

• cos²x

• sin²x

• cot²x

• tan²x

Now you know the basics!

• Tips for success:

• Memorize the identities so you can recall them quickly!

• Don't give up! There may be more than one way to simplify.

• Keep practicing! You got this!