Solving for $\theta$: Which Value is a Counterexample?

In summary, the statement "sin^2$\theta$+cos^2$\theta$=tan^2$\theta$ is not an identity because it does not hold true for all values of $\theta$. When substituting values for $\theta$, it is important to remember that $\sin^2\left({x}\right)+\cos^2\left({x}\right)=1$ and not $\tan^2\left({x}\right)=1$.
  • #1
eleventhxhour
74
0
Which value for $\theta$ is a counterexample to sin^2$\theta$+cos^2$\theta$=tan^2$\theta$ as an identity?

a) pi/4
b) 5pi/4
c) pi/3
d) It is an identity

So I tried subbing in each value (a, b, c) in as x and then finding the exact value from that but I'm not getting it.
 
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  • #2
Well, you know that $\sin^2\left({x}\right)+\cos^2\left({x}\right)=1$, so the statement you're given above is not an identity, since it does not hold true that $\tan^2\left({x}\right)=1$ for all values of $x$. :)
What can you deduce when you put the numbers on the right side, $\tan^2\left({x}\right)$? (Tongueout)
 
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Related to Solving for $\theta$: Which Value is a Counterexample?

1. What is the process for solving for $\theta$?

The process for solving for $\theta$ involves identifying the given information and using trigonometric principles and properties to manipulate equations until $\theta$ is isolated on one side of the equation.

2. How do I know if a value is a counterexample in solving for $\theta$?

A value is a counterexample if it does not satisfy the given equation or if it does not fit within the constraints of the problem. It is important to check for counterexamples in order to ensure the validity of the solution.

3. Can I use a calculator to solve for $\theta$?

Yes, a calculator can be a helpful tool in solving for $\theta$. However, it is important to understand the steps involved in solving for $\theta$ by hand in order to fully grasp the concept.

4. What are some common mistakes to avoid when solving for $\theta$?

Some common mistakes to avoid when solving for $\theta$ include not simplifying equations, not checking for extraneous solutions, and not using the correct trigonometric identities and formulas.

5. Are there any tips for solving for $\theta$ more efficiently?

Some tips for solving for $\theta$ more efficiently include practicing regularly, reviewing trigonometric principles and identities, and breaking down the problem into smaller, more manageable steps.

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