Trig Identity Homework: Solving Equations with Sin and Cos Functions

In summary, the first problem involves simplifying an expression by using the identity sin^2x + cos^2x = 1, while the second problem involves expanding and using the same identity to simplify the expression.
  • #1
tornzaer
77
0

Homework Statement


1. (sinx - cosx)(sinx + cosx) = 2sin^2x-1
2. (2sinx + 3cos)^2 + (3sinx - 2cosx)^2 = 13


Homework Equations


N/A


The Attempt at a Solution



For 1. L.S. = sinx^2+sinxcosx-sinxcosx-cosx^2, the sinxcosx cancels and I'm lost.

I haven't a clue how to do the second one.


I missed a couple of day of school because of the flu before christmas break, so I need to understand this. Some please help.
 
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  • #2
For 1. L.S. = sinx^2+sinxcosx-sinxcosx-cosx^2, the sinxcosx cancels and I'm lost.

that part is right. Now you just need sin(x)^2+cos(x)^2=1

"But what can I do with sin(x)^2-cos(x)^2?"

Nothing, but look harder. Rearrange the identity. Substitute.

Same thing with the second one, looks like you just multiply it out and substitute with that idenitity
 
  • #3
For the first one: the sinxcosx cancels out and leaves you with
[itex]sin^2 x - cos^2 x[/itex]
Then recall that [itex]sin^2 x + cos^2 x=1[/itex] from that find cos[itex]^2[/itex]x in terms of sin[itex]^2[/itex]x and sunstitute.

For the second one expand out the LHS and use the identity [itex]sin^2 x + cos^2 x=1[/itex]
 
  • #4
1. Left side is simply difference of squares, put it in (x^2-y^2) form and then use another trig identity to make it look like the right.

2. Square the terms and the middle terms with cancel the middel term of the other. From there use the fact that sin^(2)x + cos^(2)x = 1.
 
  • #5
Ok, thank you all. I understand now.
 

1. What are trigonometric identities?

Trigonometric identities are equations that involve trigonometric functions, such as sine, cosine, and tangent, and are true for all values of the variables within the domain of the functions.

2. Why is it important to learn about trigonometric identities?

Trigonometric identities are fundamental in solving trigonometric equations and in simplifying complex expressions involving trigonometric functions. They are also essential in many applications of mathematics, such as engineering, physics, and navigation.

3. How do I prove a trigonometric identity?

There are various methods for proving trigonometric identities, including using algebraic manipulations, using properties of trigonometric functions, and converting the expressions to their respective sine and cosine forms. It is important to show that the expressions are equivalent for all values of the variables within the given domain.

4. What are some common trigonometric identities?

Some common trigonometric identities include the Pythagorean identities, double angle identities, half angle identities, and sum and difference identities. These identities can be used to simplify trigonometric expressions and solve trigonometric equations.

5. How can I use trigonometric identities to solve problems?

Trigonometric identities can be used to simplify and manipulate trigonometric expressions, making them easier to solve. They can also be used to prove other mathematical statements and to solve real-world problems involving angles and distances.

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