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Is $(\sinθ)^2=\sin^2θ$

or

$(\sinθ)^2=\sin(θ^2)$

or

$(\sinθ)^2=\sin^2(θ^2)$

Can you explain your answer, regards Tom. Also, does your answer work for $\cos$ and $\tan$?

Marc van Leeuwen
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Thomas Winkworth
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    The first is correct, and the same goes for all of the trig functions. $\sin(\theta^2)$ can be more concisely written $\sin\theta^2$. And $\sin^2(\theta^2)=\sin^2\theta^2$ means $\left(\sin(\theta^2)\right)^2$, so it’s yet a third thing. – Brian M. Scott Feb 17 '15 at 02:02
  • @Brian M. Scott: The quadratic sine http://www.thefouriertransform.com/pairs/quadraticSinusoids.php#sine – orangeskid Feb 17 '15 at 07:09
  • Sine is a fuction, the first statement you made shows that the sine of a value is to be squared. If you are confused by this, then also note that $sin^{-1}x \neq \frac{1}{sinx}$ – Rohinb97 Feb 17 '15 at 14:02

7 Answers7

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$(\sin \theta)^2$ is the square of the sine of $\theta$ . Traditionally (but some people say illogically), $\sin^2 \theta$ also means this.

If you want $\sin(\theta^2)$ it is best to use parentheses.

GEdgar
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    Bravo. Yep, $sin^2\theta$ is definitely an illogical way to write $(sin \theta)^2$ and I avoid that. To me, $sin^2\theta$ is $sin(sin \theta)$. But I don't care anymore. You can't force people to write mathematics in a logical and unambiguous fashion. The notation will evolve into a myriad of stupid forms, all because of tradition. I guess if people forced each other to always be logical we'd probably speak Esperanto or Ido or some made up language. Never. gonna. happen. though. – Ray Toal Feb 17 '15 at 04:09
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$$(\sin \theta)^2=\sin^2 \theta.$$ Nothing to add

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    You could at that this is the definition of the notation "$\sin^2$", not a deep mathematical identity. – JiK Feb 17 '15 at 08:34
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Hint: $\sin^2 \theta + \cos^2 \theta =1$ means $(\sin \theta)^2 + (\cos \theta)^2 = 1$. This thing is repeated so often that it becomes $\sin^2 + \cos^2 = 1$ (for all arguments). That's one reason why $\sin^2 \theta$ is written instead of the more pedantic $(\sin \theta)^2$.

orangeskid
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In general, we write $(\sin \theta)^2$ as $\sin^2 \theta$. The same is true for cosine and tangent.

A. Thomas Yerger
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Def: $\sin^2 \theta$ = $ (\sin\theta)^2$.

If you need motivation for why we define it that way, recall the geometric definitions of sin and cos:

enter image description here

So consider the following now:

$\sin^2 \theta$ + $\cos^2 \theta$ = $(\frac {x}{r})^2$ + $(\frac {y}{r})^2$ = $(\frac {x^2 + y^2}{r^2})$ = $(\frac {r^2}{r^2})$ = 1.

This is false if we assume $ (\sin\theta^2)$ or $(\cos\theta^2)$. So this definition gives much simpler trigonometric relations.

orangeskid
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Mathemagician1234
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$(...)^2$ is a square of some number. So you first get the number then square it. $\sin(\theta^2)$ is a sine of something's square.

1

Here are some numbers to crunch

If you were to believe that

A) $(\sinθ)^2=\sin^2θ$ is true, which it is. Lets plug in $\theta=30^\circ$, so

$\sin 30^\circ=\frac{1}{2}$

Then it becomes just another value, which need be squared. So, it becomes $\frac{1}{4}$

Let's look at

B) $(\sinθ)^2=\sin(θ^2)$. If we plug in $\theta=30^\circ$, we are implying that $(\sinθ)^2=\sin (30^2)^\circ=\sin 900^\circ=0.99$

C) is an extension of B) which is just square of 0.99.

Overall, what I want to imply is these are all trigonometric ratios, they yield a certain value and they are treated like any other ratio.

MonK
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