Expression sin^n(x)/(sin^n(x)+cos^n(x))

MHB
Thread starter Vali
Start date Jan 29, 2019
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In summary, the expression sin^n(x)/(sin^n(x)+cos^n(x)) is a commonly used ratio in mathematics and physics to represent the sine and cosine functions raised to the nth power. The variable n represents the power to which the functions are raised, providing flexibility in solving equations and analyzing trigonometric patterns. This expression is related to the Pythagorean identity and the denominator, sin^n(x)+cos^n(x), allows for simplification and application in real-world scenarios such as physics, engineering, and navigation.

Jan 29, 2019

Vali

Hi!
I have sin^n(x)/(sin^n(x)+cos^n(x))
The expression is the same with 1/(ctg^n(x)+1) and I have no idea how to get to this answer.
Can you help me?

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Jan 29, 2019

MarkFL

Gold Member

MHB

If you divide the numerator and denominator by \(\sin^n(x)\) what do you get?

Jan 29, 2019

Vali

1/[(sin^n(x)+cos^n(x))/sin^n(x)]
edit:
I get it.Thanks a lot! :)

1. What does the expression sin^n(x)/(sin^n(x)+cos^n(x)) represent?

The expression represents the ratio of the sine function raised to the nth power to the sum of the sine function raised to the nth power and the cosine function raised to the nth power.

2. How do you simplify the expression sin^n(x)/(sin^n(x)+cos^n(x))?

The expression can be simplified by factoring out sin^n(x) from the numerator and denominator, leaving sin^n(x) as the common factor. This results in the simplified expression of 1/(1+cos^n(x)).

3. What is the domain of the expression sin^n(x)/(sin^n(x)+cos^n(x))?

The domain of the expression is all real numbers except for values of x that make the denominator equal to 0, which would result in an undefined expression. This occurs when x is equal to (2k+1)π/2, where k is any integer.

4. How does the value of n affect the graph of the expression sin^n(x)/(sin^n(x)+cos^n(x))?

The value of n affects the steepness of the graph. As n increases, the graph becomes steeper and approaches a vertical asymptote at x=0 and x=π/2. As n decreases, the graph becomes flatter and approaches a horizontal asymptote at y=1.

5. Can the expression sin^n(x)/(sin^n(x)+cos^n(x)) be negative?

Yes, the expression can be negative depending on the values of x and n. For example, if x=π/4 and n=2, the expression would be equal to -0.5. However, if x=π/3 and n=3, the expression would be positive. The sign of the expression depends on the values of x and n, but it is not always negative.