Precalculus Solutions | Expert Assistance for Problems & Examples

Prove this identity without using cross multiplication by manipulating one side using trig identities:
$\frac{\sin^{3} x - \cos^{3} x}{\sin x + \cos x} = \frac{\csc^{2} x - \cot x - 2 \cos^{2} x}{1 - \cot^{2} x}$
I first started off on the LHS and managed to get the denominator to become $1 - \cot^{2} x$ by multiplying by $\sin x - \cos x$ and then dividing by $\sin^{2} x$ , but from there I had no idea how to continue.

Laney Spears 2022-01-29

How do you simplify $2 (\cos (\frac{π}{2}) i \sin (\frac{π}{2}) \cdot 2 (\cos (\frac{3 π}{2}) + i \sin (\frac{3 π}{2}))$ and express the result in rectangular form?

GrareeCowui 2022-01-29

How do you simplify $6 [\cos (- \frac{π}{3}) + i \sin (- \frac{π}{3}) \cdot 3 (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))$ and express the result in rectangular form?

Duncan Reed 2022-01-29

How do I find the trigonometric form of the complex number $3 - 3 \sqrt{3} i$ .

Selena Cowan 2022-01-29

How do you write the trigonometric form of $- 1 + \sqrt{3} i$ ?

Kamren Franco 2022-01-29

How do you express the complex number in rectangular form: $12 ({\cos 60}^{\circ} + i {\sin 60}^{\circ})$ ?

Cassarrim1 2022-01-29

How do you express the complex number in rectangular form: $z = 5 ({\cos 120}^{\circ} + i {\sin 120}^{\circ})$ ?

Gavin Frye 2022-01-29

Range of $f (x) = \frac{\tan {x}}{\tan {3 x}}$
My Attempt:
I wrote down,
$\tan {3 x} = \frac{3 \tan {x} - \tan^{3} {x}}{1 - 3 \tan^{2} {x}}$
This reduced f(x) to,
$f (x) = \frac{1 - 3 \tan^{2} {x}}{3 - \tan^{2} {x}}$
I don't know how to solve any further. I thought of using derivative, but the function is dicontinuous at times.

Dayton Burnett 2022-01-29

Trig function bounded on interval (without calculus), prove that $x^{\frac{3}{2}} \sin x + \sqrt{9 - x^{3}} \cos x \leq 3$

Jaylene Franco 2022-01-29

Evaluate $\int_{| z | = 4} \tan z d z$
$\tan z = \frac{\sin z}{\cos z}$ there is a a simple pole at $z = \frac{π}{2}$
$R e s (f, \frac{π}{2}) = lim_{z \to \frac{π}{2}} (z - \frac{π}{2}) (\frac{\sin z}{\cos z})$
$R e s (f, \frac{3 π}{2}) = lim_{z \to \frac{3 π}{2}} (z - \frac{3 π}{2}) (\frac{\sin z}{\cos z})$
How to continue ?

veceriraby 2022-01-29

Denote by $V_{(x, y)}$ the vertex at ordered pair (x,y) in the Cartesian coordinate system. Denote by $A_{(x, y)}$ the measure of angle $∠ V_{(x, y)} V_{(0, 0)} V_{(1, 0)}$ . Let
$θ = \sum_{0 \leq i, j \leq 5 (i, j) \neq (0, 0)} A_{(i, j)}$
Find $θ (mod 2 π)$
I understand the question and everything, but I am slightly overwhelmed about how to find an organized approach to computing the sum of angles. I got the value down to the sum
$\sum_{0 \leq i, j \leq 5 (i, j) \neq (0, 0)} \arctan \frac{i}{j}$
But I don't know how to evaluate this. $\arctan (x + y)$ formula fails. I think there is a different approach I am not aware of.
Is there a nice way to generalize to
$\sum_{0 \leq i, j \leq k (i, j) \neq (0, 0)} A_{(i, j)}$

Haialarmz6 2022-01-29

I have an impulse train given by
$\frac{1}{R + 1} + \frac{\sum_{k = 1}^{R} \cos (\frac{2 k π x}{R + 1})}{R + 1}$
It seems obvious to me that, for x=0, the function returns 1. This is because $\cos (0) = 1$ , and we therefore end up with $\frac{1}{R + 1} + \frac{R}{R + 1} = \frac{R + 1}{R + 1} = 1$
However indeterminate result at x=0. Usually this means there is a division by 0 somewhere. But I can't see any reason for this function to produce an indeterminate result.

2alr8w 2022-01-28

Simplifying $(\sin^{6} x + \cos^{6} x) - (\sin^{4} x + \cos^{4} x) + \sin^{2} x \cos^{2} x$

Caleb Snyder 2022-01-28

Compute the limit or show it doesn't exist: $lim_{n \to \infty} (\sin \sqrt{n + 1} - \sin \sqrt{n})$

Marissa English 2022-01-28

Proving $1 - \frac{2 ϑ}{π} \sin ϑ \leq \cos ϑ$ , for $ϑ \in [0, \frac{π}{2}]$

jelentetvq 2022-01-28

Simplification of $\cos^{4} (x) + \sin^{4} (x)$
${(\sin x)}^{4} + {(\cos x)}^{4} = \frac{{(1 - \cos 2 x)}^{2}}{4} + \frac{{(1 + \cos 2 x)}^{2}}{4}$
$= \frac{1 - 2 \cos 2 x + {(\cos 2 x)}^{2} + 1 + 2 \cos 2 x + {(\cos 2 x)}^{2}}{4}$
$= \frac{1 + {(\cos 2 x)}^{2}}{2}$
its correct?

Kayleigh Dixon 2022-01-28

Help to find mistake
PSK $\lim_{x \to 0} \frac{\tan x - \sin x}{\sin^{3} x} = [\frac{1 - 1}{0}] = [\frac{0}{0}] = \lim_{x \to 0} \frac{\frac{\tan x}{x} x - \frac{\sin x}{x} x}{\sin^{3} x} = \lim_{x \to 0} \frac{x (\frac{\tan x}{x} - \frac{\sin x}{x})}{\sin^{3} x}$ ZSK
What to do with the denominator. Also the numerator in its current state converges to:
$[0 (1 - 1)] = [0]$
which is problematic since correct solution is $\frac{1}{2}$

Clare Baldwin 2022-01-28

Finding the graph defined by $x = \sin θ$ and $y = 3 - 2 \cos (2 θ)$
We are supposed to use the identity that
$\sin^{2} θ + \cos^{2} θ = 1$

blitzbabeiy 2022-01-28

Integral of $\frac{x^{2}}{\sqrt{x^{2} + 5}}$

Addison Gross 2022-01-28

Solve $\frac{d x}{d y} + \frac{x}{\sqrt{x^{2} + y^{2}}} = y$

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