Precalculus Solutions | Expert Assistance for Problems & Examples

Solving for x in $\sin^{- 1} (2 x) + \sin^{- 1} (3 x) = \frac{π}{4}$
I tried solving by differentiating both sides, but I get x=0.
How do you solve it, purely using trigonometric techniques?

Shamar Padilla 2022-01-26

The challenge trignometry question is: simplify $\sin (80^{\circ}) + \sin (40^{\circ})$ using trignometric identities. All the angle values are in degrees.
This is what I did: Let $a = 40^{\circ}$ . So we have $\sin (2 a) + \sin (a)$ . Using the formula for $\sin (2 a)$ , I have,
$2 \sin (a) \cos (a) + \sin (a) = \sin (40) (2 \cos (40^{\circ}) + 1)$
All I got was a big red mark through the question. I looked at co-functions and half-angles and couldn't find a solution that came out to something I could do without a calculator. Any thoughts?

Gavin Frye 2022-01-26

Values of x satisfying $\sin x \cdot \cos^{3} x > \sin^{3} x \cdot \cos x$
My Attempt
$\sin x \cos x \cdot (\cos^{2} x - \sin^{2} x) = \frac{1}{2} \cdot \sin 2 x \cdot \cos 2 x = \frac{1}{4} \cdot \sin 4 x > 0 ⟹ \sin 4 x > 0 x \in (0, π) ⟹ 4 x \in (0, 4 π)$
$4 x \in (0, π) \cup (2 π, 3 π) \Rightarrow x \in (0, \frac{π}{4}) \cup (\frac{π}{2}, \frac{3 π}{4})$
But, my reference gives the solution, $x \in (0, \frac{π}{4}) \cup (\frac{3 π}{4}, π)$ where am I going wrong with my attempt?

Aaden Lam 2022-01-26

Given $\tan α = 2$ , evaluate $\frac{\sin^{3} α - 2 \cos^{3} α + 3 \cos α}{3 \sin α + 2 \cos α}$

siutulysr5 2022-01-26

Given $y = A \cos (k t) + B \sin (k t)$ where A,B,k are constants, prove that $y^{(2)} + (k^{2}) y = 0$
There was a silly typo, $y^{n}$ is actually y′′

Rudy Koch 2022-01-26

How to rewrite $\cos 2 x$ in terms of $\frac{1 - \sin x}{1 + \sin x}$
I had rewritten $\cos (2 x)$ in terms of $\tan (x)$
$\cos 2 x = \frac{\cos (x) - \sin (x)}{\cos (x) + \sin (x)}$

Abbie Mcgrath 2022-01-26

Find $lim_{n \to \infty} \cos (\frac{π}{4}) \cos (\frac{π}{8}) \dots \cos (\frac{π}{2^{n}})$

minikim38 2022-01-26

Evaluating the integral $\int_{0}^{\frac{π}{4}} \sqrt{1 - 16 \sin^{2} x} d x$
I tried a substitution
$u = 4 \sin x, d x = \frac{d u}{\sqrt{16 - u^{2}}}$
hence the integration will be
$\int_{u = 0}^{u = 2 \sqrt{2}} \frac{\sqrt{1 - u^{2}}}{\sqrt{16 - u^{2}}} d u$
But I could not complete the solution using this substitution.

Jenny Branch 2022-01-26

How to solve the equation $2 z^{2} + i = - 2$ ?
Assuming that $k = - 1 - \frac{i}{2}$ we have $| k | = \frac{\sqrt{5}}{2}$ and
$\cos ϕ = \frac{- 2 \sqrt{5}}{5}, \sin ϕ = - \frac{\sqrt{5}}{5}$
How should I find $ϕ$ ?

Aubrey Hendricks 2022-01-26

Let $x \in R$ and let $n \in N$ then evaluate: $\sum_{k = 0}^{n} (\binom{n}{k}) \sin (x + \frac{k π}{2})$ I could only go up to breaking this sum in two parts of sinx and cosx in the following way: $\sum_{k = 0}^{n} (\binom{n}{k}) \sin (x + \frac{k π}{2}) = \sum_{k = 0}^{| n / 2 |} (- 1)^{k} (\binom{n}{2 k}) \sin x + \sum_{k = 0}^{| n / 2 |} (- 1)^{k} (\binom{n}{2 k + 1}) \cos x$ I have the intuition that $\sin (x + y) = \sin x \cos y + \sin y \cos x$ should be used at some point.

Jett Castaneda 2022-01-26

Error evaluating $lim_{x \to 0} \frac{x - \tan x}{x^{3}}$
I solved it like this,
$lim_{x \to 0} (\frac{1}{x^{2}} - \frac{\tan x}{x^{3}}) = lim_{x \to 0} (\frac{1}{x^{2}} - \frac{\tan x}{x} \cdot \frac{1}{x^{2}})$
Now using the property
$lim_{x \to 0} \frac{\tan x}{x} = 1$
we have:
$lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{x^{2}}) = 0$
Please explain my error! How can I avoid such errors?

Jessie Jenkins 2022-01-26

Integrating $\frac{1}{\sin (3 x)}$ when sovling an ODE

tripplek7i 2022-01-26

Prove the identity for $\tan 3 θ$
Using de Moivres

Zugdichte2r 2022-01-26

Find all 4 complex roots to $3 z^{4} - z^{3} + 2 z^{2} - z + 3 = 0$ using a trig substitution.
A previous part of the question made us prove that $2 \cos n θ = z^{n} + z^{- n}$ and it wants us to prove that the polynomial can be expressed as $6 \cos 2 θ - 2 \cos θ + 2 = 0$ . I'm assuming that using $6 \cos 2 θ - 2 \cos θ + 2 = 0$ we will be able to find all the roots of the equation but I'm unable to see how to get the polynomial into the trig equation and then how to solve the trig equation.

Aubrey Hendricks 2022-01-26

Precalc Trig Identity, verify: $1 + \cos (x) + \cos (2 x) = \frac{12}{+} \frac{\sin (5 \frac{x}{2})}{2 \sin (\frac{x}{2})}$

Lizeth Jackson 2022-01-26

Evaluating $\sum_{n = 1}^{\infty} \frac{\sin (n x)}{n}$ without integrating $\sum_{n = 1}^{\infty} e^{n x}$
My solution proceeds by integrating
$\sum_{n = 1}^{\infty} e^{n x} = \frac{e^{i x}}{1 - e^{i x}}$

Kasey Haley 2022-01-26

Evaluate $lim_{x \to 0} \frac{1 - \cos x \cos 2 x \dots \cos n x}{x^{2}}$
It should be
$\frac{1}{12} n (n + 1) (2 n + 1)$
but I don't know how to prove that. I am also aware that $lim_{θ \to 0} \frac{1 - \cos θ}{θ^{2}} = \frac{1}{2}$ but I don't know how to use that.

Wendy Gutierrez 2022-01-26

calculating 2 constants in a function
$M = {f \in C [0, 2 π], \int_{0}^{2 π} f (x) \sin x, d x = π, \int_{0}^{2 π} f (x) \sin 2 x, d x = 2 π}$
$a, b \in R, g \in M, g (x) = a \sin x + b \sin 2 x, x \in [0, 2 π]$
Ive

Marissa English 2022-01-26

Showing that $f (4 x) + 4 f (2 x) = 8 \cos^{4} (x) - 3$ ,where $f (x) = \cos x$

Swebaceacichegh 2022-01-26

If $\tan (x_{1}) \dots \tan (x_{n}) = 1$ for acute $x_{i}$ , then does it follow that $\cos (x_{1}) + \dots + \cos (x_{n}) \leq n \frac{\sqrt{2}}{2}$ ?

Precalculus problems and solutions are hard to find online even though this topic is quite popular among college students majoring in Engineering. Keeping this fact in mind, we decided to provide you with a list of questions and answers that will help you with various precalculus problems. Regardless if it is related to equations or any other problem, these precalculus problems and composite functions examples and solutions below will help you see the logic and compare solutions as you check your original task.

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