Tutor profile: Idan G.

Does sin^2(x) equal sin (x^2)?

No! There is an important distinction in the notation between these two functions. sin^2(x) is the same as sin(x) * sin (x). However, sin (x^2) just means the angle theta is being squared. For example, sin (4^2) = sin (16). Input sin (16) into your calculator and see if it equals sin (4) * sin (4).

Define a comma splice.

A comma splice is the use of a comma between two independent clauses (complete sentences) without the use of a preceding coordinating conjunction (FANBOYS = for, and, but, etc.).

What are imaginary numbers, and how are they defined? Provide an example.

Finding a square root is the inverse (reverse) operation of squaring. For example, if 3^2=9, we know the square root of 9 = 3 (3*3 = 9). Note that squaring involves multiplying the same number by itself. From basic arithmetic, we know that any two numbers multiplied by themselves will always produce a positive number. How so? 2*2 = 4 but (-2)*(-2) = 4 as well. So the square root of 4 = 2 and (-2). But what is the square root of (-4)? Can we multiply any two numbers by themselves and arrive at a negative number? We just showed that what we can't. So what does that leave with us? Does (-4) have a square root? It does, and that's where imaginary numbers come in. Side note: The term "imaginary numbers" was originally meant to be derogatory, but since then, they have developed important applications in electrical engineering, physics, and other fields. Since no two "real" numbers multiplied by themselves produce a negative number (1*1=1 and (-1)*(-1)=1), we define the square root of (-1) as the imaginary number "i". From here we can begin performing various operations. For example, i^2=1 ((-1)*(-1)=1). i^3=-1 ((-1)*(-1)=1 and 1*(-1)=-1), etc. We can also find negative square roots. For example, the square root of (-9) = sqrt 9 * sqrt -1 = 3 * i = 3i.