HPC Section 5.1 Part 1

Presentation

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Mathematics

•

9th - 12th Grade

•

Medium

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CCSS

HSF.TF.C.8, HSG.SRT.C.6, HSF.TF.A.2

Standards-aligned

Shane Devlin

Used 14+ times

FREE Resource

19 Slides • 28 Questions

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

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?

$\cos^3x+\sin^2x\cos x$

$\sin x$

$\cos x$

$\tan x$

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!)

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

$\csc x-\sec x$

$\sec x-\csc x$

PreCalculus
Section 5.1 Part 1

Using Fundamental Identities

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HPC Section 5.1 Part 1

PreCalculus Section 5.1 Part 1

Identity Categories:

Reciprocal Identities:

Quotient Identities:

Pythagorean identity #1:

Pythagorean Identity #2

Pythagorean Identity #3

Simplifying Suggestions

Homework

Now you know the basics!

PreCalculus Section 5.1 Part 1

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Section 5.1 Part 1

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