HPC Section 5.1 Part 1

PreCalculus
Section 5.1 Part 1

​Using Fundamental Identities

Multiple Select

Given the diagram, which of the following statements are true?

• $$x^2+y^2=r^2$$

• $$\sin\theta=\frac{y}{r}$$

• $$r=x+y$$

• $$\sin^2\theta+\cos^2\theta=1$$

• $$x=\sqrt{r^2-y^2}$$

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!

Multiple Choice

 Which of the following would be the definition of

$$\cos\theta$$  ?

• $$\frac{x}{y}$$  

• $$\frac{x}{r}$$  

• $$\frac{y}{r}$$  

• $$\frac{r}{x}$$  

• $$\frac{r}{y}$$  

Multiple Choice

 Which of the following would be the definition of
$$\cot\theta$$  ?

• $$\frac{x}{y}$$  

• $$\frac{x}{r}$$  

• $$\frac{y}{r}$$  

• $$\frac{r}{x}$$  

• $$\frac{r}{y}$$  

Multiple Choice

 Which of the following would be the definition of
$$\sec\theta$$  ?

• $$\frac{x}{y}$$  

• $$\frac{x}{r}$$  

• $$\frac{y}{r}$$  

• $$\frac{r}{x}$$  

• $$\frac{r}{y}$$  

Multiple Choice

 Which of the following would be the definition of
$$\csc\theta$$  ?

• $$\frac{x}{y}$$  

• $$\frac{x}{r}$$  

• $$\frac{y}{r}$$  

• $$\frac{r}{x}$$  

• $$\frac{r}{y}$$  

Reciprocal Identities: 

Quotient Identities:

Multiple Choice

*

• sin x
• cos x
• csc x
• cot x

Multiple Choice

*

• csc x
• sec x
• cot x
• tan x

Multiple Choice

*

• cos x
• csc x
• sec x
• cot x

Multiple Choice

*

• csc x
• sec x
• cot x
• tan x

Multiple Choice

*

• csc x
• sec x
• cot x
• tan x

Multiple Choice

Use

$$\csc x=2$$

find the other trig functions.  sin x = ?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{2\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• $$\frac{\sqrt{3}}{3}$$  

Multiple Choice

Use

$$\csc x=2$$  

find the other trig functions.  If x is in the first quadrant what is cos x = ?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{2\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• $$\frac{\sqrt{3}}{3}$$  

Multiple Choice

Use

$$\csc x=2$$  

find the other trig functions.  If x is in the first quadrant what is sec x = ?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{2\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• $$\frac{\sqrt{3}}{3}$$  

Multiple Choice

Use

$$\csc x=2$$

 find the other trig functions.  If x is in the first quadrant what is tan x = ?

• $$\frac{1}{2}$$  

• $$\frac{\sqrt{3}}{2}$$  

• $$\frac{2\sqrt{3}}{3}$$  

• $$\sqrt{3}$$  

• $$\frac{\sqrt{3}}{3}$$  

Pythagorean identity #1:

Pythagorean Identity #2

Pythagorean Identity #3

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

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

Multiple Select

$$\sin^2\theta+\cos^2\theta=1$$  

Choose the equivalent equations?

• $$\sin^2\theta=1-\cos^2\theta$$  

• $$\cos^2\theta=1-\sin^2\theta$$  

• $$\sin^{ }\theta=\sqrt{1-\cos^2\theta}$$  

• $$\cos\theta=\sqrt{1-\sin^2\theta}$$  

• $$\sin\theta=1-\cos\theta$$  

Multiple Select

$$1+\ \tan^2\theta=\sec^2\theta$$  

Choose the equivalent equations?

• $$\tan^2\theta=\sec^2\theta-1$$  

• $$\tan^2\theta-\sec^2\theta=1$$  

• $$\sec^2\theta-\tan^2\theta=1$$  

• $$\tan\theta=\sqrt{\sec^2\theta-1}$$  

• $$\sec\theta=1+\tan\theta$$  

Simplifying trigonometric expressions.

​Using your resources

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\$$  

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

Homework

Day 1
Page 354 – 355 # 1, 2, 7, 11, 13, 17, 21 – 25, 33 – 39 odd

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:
$$\sin x\left(\sec x-\csc x\right)$$

• $$1-\tan x$$  

• $$\tan x-1$$  

• $$1-\cot x$$  

• $$\sin x$$  

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!

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

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

• $$\sec x-\csc x$$