all trigonometric identities double angle formula

ALL TRIGONOMETRIC IDENTITIES DOUBLE ANGLE FORMULA HOW TO
ALL TRIGONOMETRIC IDENTITIES DOUBLE ANGLE FORMULA PLUS

We should use some tools in our toolkit, one of which We might as well give up, but we have this indeterminateįorm, it does not mean the limit does not exist. Non-zero divided by zero, we'd say, okay, that's undefined. Just going to be zero, so what we end up with is equal to zero over zero, and as we've talked about before, if we had something All of this stuff simplifies to negative 1 over, this is going to be cosine of negative pi over 2, right? This is negative pi over 2, cosine of negative pi overĢ, if you thought in degrees, that's going to be negative 90 degrees.

ALL TRIGONOMETRIC IDENTITIES DOUBLE ANGLE FORMULA PLUS

That it's good to know and so if you have, if you have 1, so let's see, actually, let me just rewrite it, so this is going to be equal to 1 plus square root of 2 times that is going to be negative 2 over 2, so this is going to be minus 1, that's the numerator over here. So this is one of the, one of the trig values Square root of 2 over 2, we're assuming this is in radians, if we're thinking inĭegrees, this would be a negative 45-degree angle, Now, negative pi over 4, sine of negative pi over 4 is going to be negative square root of 2 over 2, so this is negative Going to be equal to this expression evaluatedĪt negative pi over 4, so 1 plus square root of 2 times sine of negative pi over 4, over cosine of 2 times negative pi over 4. Make sure we can see that negative there, of cosine of 2 theta, and both of these expressions are, if these were function definitions or if we were to graph y equals 1 plus square root of sine, square root of 2 times sine theta, or y equals cosine of 2 theta, we would get continuous functions, especially at theta isĮqual to negative pi over 4, so we could just substitute in. Going to be the same thing as the limit, as theta approaches negative pi over 4 of 1 plus square root of 2 sine theta over the limit as theta approaches negative pi over 4. Well, one take on it is well, let's just, let's just say that this is It a shot before we go through it together. Root of 2 sine of theta over cosine of 2 theta, as theta approaches negative pi over 4.

See if we can find the limit of 1 over the square In other words, the discontinuity doesn’t really exist, it only seems to exist due to the equation not being in its simplest, i.e. If a function can be factored, and the factoring removes a discontinuity, it seems more logical to me to state that the original function had a false or phantom discontinuity.

ALL TRIGONOMETRIC IDENTITIES DOUBLE ANGLE FORMULA HOW TO

I am comfortable performing such manipulations, as it makes inherent sense to me that the limits are equivalent but I wouldn’t know how to mathematically state or prove that such manipulations are correct. I believe mathematicians refer to such a discontinuity as a ‘removable discontinuity’. where a factored equation has removed a discontinuity which was present in the original unfactored equation, and that the limits are considered equivalent. I want to better understand this distinction as Sal has identified this situation several times in multiple videos, i.e. For I learned from the video on Epsilon-Delta proofs that a limit can still exists at 'a' despite a function being undefined at 'a'. Is there a mathematical theorem or proof that states the limit of a factored equation is equivalent to the limit of the respective unfactored equation as long as the factored equation is continuous at the point for which the limit is taken? Is this distinction covered by the Epsilon-Delta proof of limits. Sal goes on to say that the limits of f(x) and g(x) are equivalent at 'a'. And that f(x) is defined and is continuous at ‘a’. I'm assuming this means that g(x) is defined for all real numbers except 'a', and therefore is not continuous at 'a'. In the video, Sal states that f(x) and g(x) are equivalent except at ‘a’, and that f(x) is continuous at ‘a’.

I’m assuming that g(x) is the original unfactored equation and that f(x) is the resulting factored equation. In his explanation he speaks of two functions f(x) and g(x).

8:20 in the video, Sal begins to explain that it is important to understand that the factored equation is not the same as the original unfactored equation.

All trigonometric identities double angle formula