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5.1 Using Fundamental Identities

Integrated Math 3 Honors

Mr. Washington

Objectives

• Recognize and write the fundamental trigonometric identities.

• Use the fundamental trigonometric identities to evaluate trigonometric functions, simplify trigonometric expressions, and rewrite trigonometric expressions. 

Uses for Trigonometric Identities

• To Evaluate Trig Functions

• Simplify Trig Expressions

• To Develop Additional Trig Functions

• To Solve Trig Equations

Fundamental Trigonometric Identities

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Example 1

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Example 2

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Example 3

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Example 4

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Example 5

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Example 6

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Multiple Choice

Simplify

 $\csc^2x\left(1-\cos^2x\right)$  

• 1

• Already simplified

• -1

Solution

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Multiple Choice

 $\frac{\cot^2x}{\csc^2x}$  Simplify

•  $\cot^2x$  

•  $\sin^2x$  

•  $\cos^2x$  

•  $\sec^2x$  

Solution

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Multiple Choice

 $\frac{1}{1+\cos^2\theta}$  

•  $\sin^2x$  

•  $\cos^2x$  

•  $\sec^2x$  

Solution

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Multiple Choice

 $\csc\ t\ +\ \tan\ t\ +\sec t$  

• 1 sec t

• 2 sec t

• 3 sec t

• 4 sec t

Solution

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Multiple Choice

sin² Θ + cos² Θ =

• 1

• 0

• -1

• tan² Θ

Multiple Choice

• tan Θ

• cot Θ

• 1

• none of these

Multiple Choice

• cos Θ

• csc Θ

• sec Θ

• none of these

Multiple Choice

• sin Θ

• csc Θ

• sec Θ

• none of these

Multiple Choice

• sin Θ

• cos Θ

• csc Θ

• none of these

Multiple Choice

• sin Θ

• cos Θ

• sec Θ

• none of these

Multiple Choice

1 + tan² Θ =

• sec² Θ

• csc² Θ

• cot² Θ

• none of these

Multiple Choice

cot² Θ + 1 =

• sec² Θ

• csc² Θ

• tan² Θ

• none of these

Multiple Select

Check all that are true.

• sin x = sin (-x)

• -csc x = csc (-x)

• cos (-x) = cos x

• -tan x = tan (-x)

• sec x = sec (-x)

Multiple Choice

sin² Θ + cos² Θ =

• 1

• 0

• -1

• tan² Θ

Multiple Choice

• tan Θ

• cot Θ

• 1

• none of these

Multiple Choice

• cos Θ

• csc Θ

• sec Θ

• none of these

Multiple Choice

• sin Θ

• csc Θ

• sec Θ

• none of these

Multiple Choice

• sin Θ

• cos Θ

• csc Θ

• none of these

Multiple Choice

• sin Θ

• cos Θ

• sec Θ

• none of these

Multiple Choice

1 + tan² Θ =

• sec² Θ

• csc² Θ

• cot² Θ

• none of these

Multiple Choice

cot² Θ + 1 =

• sec² Θ

• csc² Θ

• tan² Θ

• none of these

Multiple Choice

Which of the following is not a pythagorean identity?

• cot²x + 1 = csc²x

• csc²x + 1 = cot²x

• cot²x - csc²x = -1

• csc²x - cot²x = 1

Multiple Select

Select all that are true.

• $\cot\theta=\tan\left(\frac{\pi}{2}-\theta\right)$

• $\sec\left(\frac{\pi}{2}-\theta\right)=\csc\theta$

• $\sin\theta=\cos\ \left(\frac{\pi}{2}-\theta\right)$

• $\cot\left(\frac{\pi}{2}-\theta\right)=\tan\theta$

• $\csc\left(\frac{\pi}{2}-\theta\right)=\sec\theta$

Multiple Select

Check all that are true.

• sin x = sin (-x)

• -csc x = csc (-x)

• cos (-x) = cos x

• -tan x = tan (-x)

• sec x = sec (-x)

Using Fundamental Identities

Lesson 5.1

Pre Calculus

Factoring Review (3 terms)

If you need a review of factoring 3 term polynomials please watch the video on the next slide. You might have learned to factor using the "X method", "Guess and check" or another method - any method is fine as long as you are getting the correct answer. If you do not remember how to factor you should walk the factoring review video on the next slide!

Multiple Choice

Factor the expression $$x^2-x-2$$

• $$\left(x-1\right)\left(x-1\right)$$

• $$\left(x-2\right)\left(x-1\right)$$

• $$\left(x+1\right)\left(x-1\right)$$

• $$\left(x-2\right)\left(x+1\right)$$

Multiple Choice

Factor the expression $$4x^2+x-3$$

• $$\left(x-3\right)\left(x+4\right)$$

• $$\left(2x-3\right)\left(2x+1\right)$$

• $$\left(x+1\right)\left(4x-3\right)$$

• $$\left(x-3\right)\left(x+1\right)$$

Multiple Choice

Factor the expression $$4x^2+x-3$$

• $$\left(4x-3\right)\left(x+1\right)$$

• $$\left(x-3\right)\left(x-2\right)$$

• $$\left(2x+3\right)\left(x+2\right)$$

• $$\left(x-2\right)\left(2x+3\right)$$

Factoring Review (Difference of 2 squares)

If you need a review of factoring by a difference of 2 squares please watch the video on the next slide. If you do not remember how to factor you should walk the factoring review video on the next slide!

Multiple Choice

Factor the expression $$x^2-1$$

• $$\left(x+1\right)\left(x+1\right)$$

• $$\left(x+1\right)\left(x-1\right)$$

• $$\left(x-1\right)\left(x-1\right)$$

• $$x-1$$

Multiple Choice

Factor the expression $$1-4x^2$$

• $$\left(2x+1\right)\left(2x-1\right)$$

• $$\left(1-2x\right)\left(1+2x\right)$$

• $$\left(4x-1\right)\left(x+1\right)$$

• $$\left(1-x\right)\left(1+4x\right)$$

Multiple Choice

Factor out the GCF: $$xy^2-x$$

• $$x\left(y^2-x\right)$$

• $$x\left(y-1\right)$$

• $$x\left(y^2-1\right)$$

• $$xy\left(y-1\right)$$

Multiple Choice

Factor out the GCF: $$\sin\left(x\right)\cos^2\left(x\right)-\sin\left(x\right)$$

• $$\sin\left(x\right)\left[\cos^2\left(x\right)-\sin\left(x\right)\right]$$

• $$\sin\left(x\right)\left[\cos^2\left(x\right)-1\right]$$

• $$\sin\left(x\right)\cos\left(x\right)\left[\cos\left(x\right)+1\right]$$

​You will use the pythagorean identities throughout these notes!

​You might need to rearrange the identities to use them. Some ways to rearrange them are below: