How do you determine the restrictions of an identity?

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  • Thread starter eleventhxhour
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So, in summary, the restrictions on the variables for this identity are that the angle cannot be 0°, 90°, 180°, 270°, or 360°.
  • #1
eleventhxhour
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So, this question says "prove each identity. State any restrictions on the variables".

5a) \(\displaystyle \frac{sinx}{tanx} = cosx\)

I did the first part of the question correctly (proving it), but I don't understand how you determine the restrictions on the variables. In the textbook, it says that it cannot be 0°, 90°, 180°, 270°, and 360°. Could someone explain how they got that?

Thanks!
 
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  • #2
We cannot have division by zero, so we know:

\(\displaystyle \tan(x)\ne0\)

And because \(\displaystyle \tan(x)\equiv\frac{\sin(x)}{\cos(x)}\implies\sin(x)\ne0\)

We also have:

\(\displaystyle \cos(x)\ne0\)
 
  • #3
MarkFL said:
We cannot have division by zero, so we know:

\(\displaystyle \tan(x)\ne0\)

And because \(\displaystyle \tan(x)\equiv\frac{\sin(x)}{\cos(x)}\implies\sin(x)\ne0\)

We also have:

\(\displaystyle \cos(x)\ne0\)

Okay, so that gets you that it cannot = 0° and 90°. But how did they get that it cannot be 180°, 270°, and 360°?
 
  • #4
What are:

\(\displaystyle \sin\left(180^{\circ}\right)\)

\(\displaystyle \cos\left(270^{\circ}\right)\)

\(\displaystyle \sin\left(360^{\circ}\right)\)
 
  • #5
MarkFL said:
What are:

\(\displaystyle \sin\left(180^{\circ}\right)\)

\(\displaystyle \cos\left(270^{\circ}\right)\)

\(\displaystyle \sin\left(360^{\circ}\right)\)

They all equal 0
 

FAQ: How do you determine the restrictions of an identity?

1. How do you determine the restrictions of an identity?

The restrictions of an identity can be determined by examining the domain and range of the functions involved in the identity. Any values that result in undefined or impossible outputs should be considered as restrictions.

2. What are some common restrictions that can be found in identities?

Common restrictions in identities include division by zero, taking the square root of a negative number, and logarithms of non-positive numbers. These restrictions are usually found in expressions involving fractions, radicals, and logarithmic functions.

3. Can restrictions vary depending on the type of identity?

Yes, the restrictions of an identity can vary depending on the type of functions involved. For example, trigonometric identities may have restrictions related to the domain of trigonometric functions, while algebraic identities may have restrictions related to the domain and range of algebraic functions.

4. How can restrictions affect the validity of an identity?

If an identity has restrictions, it means that there are certain values that cannot be substituted into the expression, making it invalid for those values. In order for an identity to be valid, it must hold true for all possible values within its domain.

5. Are restrictions the same as limitations in identities?

No, restrictions are not the same as limitations in identities. Restrictions refer to specific values that cannot be used in an identity, while limitations refer to the overall scope or applicability of the identity. Restrictions can be seen as technical limitations, while limitations are more conceptual.

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