[25sin^2(x)+9cos^2(x)]=[9+16sin^2(x)] Why?

  • Thread starter The Sand Man
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In summary, the simplification of [25sin^2(x)+9cos^2(x)] to [9+16sin^2(x)] is based on the fact that [ sin^2(x) + cos^2(x) = 1 ]. By writing the equation as [ (16+9)sin^2(x) + 9cos^2(x) ] and using the property [ sin^2(x) + cos^2(x) = 1 ], we can simplify [16sin^2(x) + 9sin^2(x) + 9cos^2(x) ] to [16sin^2(x) + 9(1) ] and finally to [ 16 sin^2(x
  • #1
The Sand Man
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I'm doing a problem where I need to find the maximum curvature. I'm at a point where I need to simplify the denominator and reduce the funcitions in the bottom. I don't understand how to simplify:

[25sin^2(x)+9cos^2(x)]

To:

[9+16sin^2(x)]

What is getting factored out? Or is this completing the square somehow?

Thanks for any help.
 
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  • #2
the simplyfication is based on the fact [ sin^2(x) + cos^2(x) = 1 ]

write down ur equation as: [ (16+9)sin^2(x) + 9cos^2(x) ] ====
[ 16sin^2(x) + 9sin^2(x) + 9cos^2(x) ] =====

[ 16sin^2(x) + 9{ sin^2(x) + cos^2(x) } ] ====

[ 16sin^2(x) + 9(1) ] ====

[ 16 sin^2(x) + 9 ]
 
  • #3
abluphoton said:
the simplyfication is based on the fact [ sin^2(x) + cos^2(x) = 1 ]

write down ur equation as: [ (16+9)sin^2(x) + 9cos^2(x) ] ====
[ 16sin^2(x) + 9sin^2(x) + 9cos^2(x) ] =====

[ 16sin^2(x) + 9{ sin^2(x) + cos^2(x) } ] ====

[ 16sin^2(x) + 9(1) ] ====

[ 16 sin^2(x) + 9 ]

Yeah I'm retarded. I knew it would be something that simple. Thanks
 

1. Why is [25sin^2(x)+9cos^2(x)]=[9+16sin^2(x)]?

The equation [25sin^2(x)+9cos^2(x)]=[9+16sin^2(x)] is a result of applying the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. By substituting cos^2(x) = 1-sin^2(x) in the original equation, we get [25sin^2(x)+9(1-sin^2(x))]=[9+16sin^2(x)]. After simplifying, we get the same equation on both sides.

2. How is the Pythagorean identity used in this equation?

The Pythagorean identity is used to simplify the original equation and make it easier to solve. By using the identity, we can eliminate one of the variables (cos^2(x)) and reduce the equation to only one variable (sin^2(x)). This makes it easier to find the solutions for x.

3. Can this equation be solved for x?

Yes, this equation can be solved for x. By simplifying the equation using the Pythagorean identity and rearranging the terms, we can get a quadratic equation in terms of sin^2(x). This can be solved using the quadratic formula to find the solutions for x.

4. What are the solutions for x in this equation?

The solutions for x in this equation depend on the values of sin^2(x). Since sin^2(x) can take on any value between 0 and 1, the solutions for x can vary. However, by solving the quadratic equation obtained from the simplified equation, we can find the specific values of sin^2(x) that satisfy the equation and correspond to the solutions for x.

5. How does this equation relate to trigonometric functions?

This equation relates to trigonometric functions because it involves the use of trigonometric identities (specifically, the Pythagorean identity) to simplify and solve the equation. The equation also involves the use of sine and cosine functions, which are fundamental trigonometric functions.

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