Sin^2(x) + tan^2(x) = √2

In summary, the student attempted to solve for sin2(x) using various trigonometric identities but was stuck at the last step.
  • #1
47
6

Homework Statement


The problem given is sin2(x) + tan2(x) = √2

2. Homework Equations


The relevant equations would be any trigonometric identities

The Attempt at a Solution



sin2(x) + tan2(x) = √2

sin2(x) + (sin2(x)/cos2(x) ) = √2

[ cos2(x) sin2(x) + sin2(x) ]/ cos2(x) = √2

[ (1- sin2(x)) sin2(x) + sin2(x)] / [ 1 - sin2(x) ] = √2

[ (2 - sin2(x) ) sin2(x) ] / [ 1 - sin2(x) ] = √2

(2 - sin2(x) ) tan2(x) = √2

tan2(x) = ( √2 / [ 2 - sin2(x) ] )

I take this and substitute into the first equation:

sin2(x) + ( √2 / [ 2 - sin2(x) ] ) = √2

( 2 - sin2(x) ) sin2(x) / [ 2 - sin2(x) ] + √2 / [ 2 - sin2(x) ] = √2

( 2 - sin2(x) ) sin2(x) + √2 / [ 2 - sin2(x) ] = √2

( 2 - sin2(x) ) sin2(x) + √2 = ( √2 ) 2 - sin2(x)

( 2 - sin2(x) ) sin2(x) + √2 = 2√2 - √2 sin2(x)

( 2 - sin2(x) ) sin2(x) = √2 - √2 sin2(x)

( 2 - sin2(x) ) sin2(x) = √2 (1 - sin2(x) )

( ( 2 - sin2(x) ) sin2(x) / (1 - sin2(x) ) )= √2Here is where I get stuck. I do not know what steps to take next. Please give me hints on this and do not hesitate to point out any mistakes in my work. They are very likely.
 
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  • #2
ForceBoy said:

Homework Statement


The problem given is sin2(x) + tan2(x) = √2

2. Homework Equations


The relevant equations would be any trigonometric identities

The Attempt at a Solution



sin2(x) + tan2(x) = √2

sin2(x) + (sin2(x)/cos2(x) ) = √2

[ cos2(x) sin2(x) + sin2(x) ]/ cos2(x) = √2

[ (1- sin2(x)) sin2(x) + sin2(x)] / [ 1 - sin2(x) ] = √2

[ (2 - sin2(x) ) sin2(x) ] / [ 1 - sin2(x) ] = √2
I can follow you until here. What's next is a backward substitution which I don't think will get you very far. At least the next steps are what could be done more easily, because you already have a quadratic equation in ##t := \sin^2(x)## which can be solved.
(2 - sin2(x) ) tan2(x) = √2

tan2(x) = ( √2 / [ 2 - sin2(x) ] )

I take this and substitute into the first equation:

sin2(x) + ( √2 / [ 2 - sin2(x) ] ) = √2

( 2 - sin2(x) ) sin2(x) / [ 2 - sin2(x) ] + √2 / [ 2 - sin2(x) ] = √2

( 2 - sin2(x) ) sin2(x) + √2 / [ 2 - sin2(x) ] = √2

( 2 - sin2(x) ) sin2(x) + √2 = ( √2 ) 2 - sin2(x)

( 2 - sin2(x) ) sin2(x) + √2 = 2√2 - √2 sin2(x)

( 2 - sin2(x) ) sin2(x) = √2 - √2 sin2(x)

( 2 - sin2(x) ) sin2(x) = √2 (1 - sin2(x) )

( ( 2 - sin2(x) ) sin2(x) / (1 - sin2(x) ) )= √2Here is where I get stuck. I do not know what steps to take next. Please give me hints on this and do not hesitate to point out any mistakes in my work. They are very likely.
 
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  • #3
Wow, I didn't see that! Thank you very much. This was really helpful.
 

1. What is the significance of the equation Sin^2(x) + tan^2(x) = √2 in mathematics?

This equation is known as a trigonometric identity, meaning it is true for all values of x. It is often used in trigonometry and calculus to simplify expressions and solve equations.

2. Can this equation be proved using mathematical principles?

Yes, the equation can be proven using the Pythagorean identity, which states that sin^2(x) + cos^2(x) = 1. By dividing both sides by cos^2(x), we get the equation sin^2(x)/cos^2(x) + 1 = sec^2(x). Since sec^2(x) = 1 + tan^2(x), we can substitute in the original equation to get sin^2(x)/cos^2(x) + tan^2(x) = 1 + tan^2(x). Simplifying this, we get sin^2(x) + tan^2(x) = √2.

3. How is this equation used in real-world applications?

This equation is commonly used in physics and engineering, specifically in fields that involve wave motion and oscillation. It can also be used in navigation and surveying to calculate the distance between two points.

4. Are there any other forms of this equation?

Yes, there are several other forms of this equation, including Sin^2(x) + cot^2(x) = csc^2(x) and cos^2(x) + cot^2(x) = 1. These can be derived from the Pythagorean identity and used in similar applications.

5. Is there a specific method to solve equations involving this trigonometric identity?

To solve equations such as sin^2(x) + tan^2(x) = √2, you can use algebraic techniques to manipulate the equation and isolate the variable. You can also use a graphing calculator to find the intersection points of the two sides of the equation.

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