Simplifying a Trig Expression

In summary, the conversation was about a person who was bored and decided to review their old math books. They were working on a problem involving trigonometric identities and made some progress before getting stuck. They received a hint from someone else which triggered a realization and they were able to solve the problem. They also mentioned the need for more frequent reviews of algebra and trigonometry.
  • #1
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The other day in a fit of boredom I decided to dust off my old math books (high school and undergrad) and see if I can still crank through the basics.

1. Homework Statement

Show the following: $$ \frac { 3 cos^2(x) } { 2 - 2 sin(x) } = \frac { 3 }{ 2 } ( sin(x) + 1 ) $$

Homework Equations


$$ sin^2(x) + cos^2(x) = 1 $$ $$ cos^2(x) = 1 - sin^2(x) $$

The Attempt at a Solution


I make it about halfway to a solution and then draw a blank.
$$ \frac { 3 cos^2(x) } { 2 - 2 sin(x) } = \frac { 3 (1 - sin^2(x)) } { 2 - 2 sin(x) } $$ $$ = \frac { 3 - 3 sin^2(x) } { 2 - 2 sin(x) } $$ I get the feeling I've overlooked something fairly obvious or swapped a sign somewhere, but it's not jumping off the page at me. Any nudges in the right direction are greatly appreciated.
 
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  • #2
Hint: ##(1+x)(1-x) = 1 - x^2##
 
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Likes Michael Price, neilparker62, YoungPhysicist and 1 other person
  • #3
I'm guessing I'd apply that before having multiplied through by 3, giving $$ \frac { 3 (1 + sin(x)) (1 - sin(x)) } { 2 - 2 sin(x) } = \frac { 3 (1 + sin(x)) (1 - sin(x)) } { 2 (1 - sin(x)) } $$ Hey! It works! The ## ( 1 - sin(x) )'s ## cancel out!
I should do a review of algebra and trig more than once a decade :-)
 
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Likes chwala, YoungPhysicist and PeroK
  • #4
DrClaude said:
Hint: ##(1+x)(1-x) = 1 - x^2##
Triggered a 'Eureka' moment!
 
  • #5
atomicpedals said:
I'm guessing I'd apply that before having multiplied through by 3, giving $$ \frac { 3 (1 + sin(x)) (1 - sin(x)) } { 2 - 2 sin(x) } = \frac { 3 (1 + sin(x)) (1 - sin(x)) } { 2 (1 - sin(x)) } $$ Hey! It works! The ## ( 1 - sin(x) )'s ## cancel out!
I should do a review of algebra and trig more than once a decade :-)

this was easy though...for a post graduate bingo...
 

1. What does it mean to simplify a trig expression?

Simplifying a trig expression means to rewrite it in a more compact or simplified form by using trigonometric identities, properties, and rules.

2. Why is it important to simplify trig expressions?

Simplifying trig expressions can help make calculations and solving equations easier and more efficient. It also allows for a better understanding of the relationship between different trig functions.

3. What are some common trigonometric identities used to simplify expressions?

Some common trigonometric identities used to simplify expressions include the Pythagorean identities, double angle identities, half angle identities, and sum and difference identities.

4. How do you know when a trig expression is fully simplified?

A trig expression is fully simplified when it cannot be further reduced or rewritten using any of the trigonometric identities or properties.

5. Are there any tips for simplifying trig expressions?

Some tips for simplifying trig expressions include looking for common factors, using the reciprocal and quotient identities, and converting all trig functions to sine and cosine. It is also helpful to practice and familiarize yourself with the various identities and properties.

Suggested for: Simplifying a Trig Expression

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