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bobsmith76
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Homework Statement
what's the difference between sin^2(x) and sin(x^2)?
Homework Equations
The Attempt at a Solution
I can't do sin^2(x) on the calculator. let's just use sin(25) as an example, what's the difference?
Indeed. if x = 5, sin2(x) ≈ 0.919536, whereas sin(x2) ≈ -0.132352bobsmith76 said:Homework Statement
what's the difference between sin^2(x) and sin(x^2)?
The Attempt at a Solution
I can't do sin^2(x) on the calculator. let's just use sin(25) as an example, what's the difference?
\( \sin^2(x) \) is a mathematical expression that represents the square of the sine of an angle \( x \). It means \( \sin(x) \) multiplied by itself. In other words, it's equivalent to \( (\sin(x))^2 \).
\( \sin(x^2) \) is a mathematical expression that represents the sine of the square of an angle \( x \). It means taking the sine of \( x \) squared, or \( \sin((x)^2) \).
The key difference lies in the order of operations. In \( \sin^2(x) \), you first find the sine of \( x \) and then square the result. In \( \sin(x^2) \), you first square \( x \) and then find the sine of the squared value.
No, the values of \( \sin^2(x) \) and \( \sin(x^2) \) are generally different. This is because squaring the sine of an angle and taking the sine of the squared angle produce different results, except in specific cases where \( \sin(x) = 0 \) or \( \sin(x) = \pm 1 \).
Certainly! Let's take an example: If \( x = 30^\circ \) (or \( \frac{\pi}{6} \) radians), then \( \sin(30^\circ) = \frac{1}{2} \). Therefore, \( \sin^2(30^\circ) = \left(\frac{1}{2}\right)^2 = \frac{1}{4} \). On the other hand, \( \sin((30^\circ)^2) = \sin(900^\circ) = \sin(2\pi) = 0 \).
Yes, \( \sin^2(x) \) is often used in trigonometry and mathematics to represent the square of the sine function, which can be used in various mathematical calculations. \( \sin(x^2) \) may appear in more complex mathematical or scientific equations where the sine of a squared value is required.
One common misconception is the incorrect use of notation. Some people may write \( \sin^2(x) \) when they mean \( (\sin(x))^2 \), and vice versa. It's important to understand the order of operations to use the correct notation and interpretation.