Precalculus Solutions | Expert Assistance for Problems & Examples

Want to do is find out how many solutions there are to the following equation in that interval (here A is a constant): $\cos (A \cos (\frac{s}{A})) = \sin (A \cos (\frac{s}{A})) = A \sin (\frac{s}{A}), s \in [0, 2 A π]$
I know it depends on A and I've tested a few cases, but I'm rusty on this trig stuff, so I haven't been able to do it yet

Keaton Good 2022-01-27

$\sin (x) - x \cos (x) - \frac{1}{2} x^{2} = 0$
I can verify that 0 is the solution of equation. But i don't have any method to show that is the only solution.

Calvin Hess 2022-01-27

How can I prove ${\tan 15}^{\circ} = 2 - \sqrt{3}$ using the triple-angle formulas?
$\sin 3 x = 3 \sin x - 4 \sin^{3} x$
$\cos 3 x = 4 \cos^{3} x - 3 \cos x$
I know that if I substitute $x = 15^{\circ}$ I can write
$\sin 45 = 3 \sin 15 - 4 \sin^{3} 15$
$\cos 45 = 4 \cos^{3} 15 - 3 \cos 15$
What can I do next?any hints

Jessie Jenkins 2022-01-27

Prove that $f (x) = \frac{\sin (x + α)}{\sin (x + β)}$ is monotonic in any interval of its domain, $α, β \in R$
Obviously $x + β \neq π n$ . Then i've tried to split the problem into three parts: $α = β, α > β, β > α$ .
For the first case it's obvious that the function turns into a constant and therefore is monotonic.
Since f(x) is periodic we may use that fact and consider some interval for example $(0, \frac{π}{2}) or (0, π)$ .
But since $α and β$ may take any values i didn't manage to figure out any inequality to start from.
What steps should i take to proceed?

Isis Solis 2022-01-27

Given:
$\sin (x) = u \sin (x + a), {u < 1}$
How do I expand x in powers of u?

Aubrey Hendricks 2022-01-27

Why does $\cos θ = \sin (\frac{π}{2} - θ)$ even when $θ > 90^{\circ}$ ?
I get why this relationship holds true in 90 degree triangles, but I don't get why this relationship holds true when $θ > 90^{\circ}$ ? I get that when $θ > 90^{\circ}$ you find the acute angle that describes $θ$ , if you see what I mean. But there is no way of showing that in this function! So why would the fact that $\cos θ = \sin (\frac{π}{2} - θ)$ still hold true here?

Yahir Haas 2022-01-27

Solve $\tan (x - \frac{π}{4}) = - \tan (x + \frac{π}{2})$

Babenzgs 2022-01-27

General Sine and Cosine formula for sum of a finite number of angles
I was wondering is there is a general formula for $\sin (x_{1} + x_{2} + x_{3} + \dots + x_{n})$ as well as for the cosine function. I know that $\sin (x_{1} + x_{2}) = \sin (x_{1}) \cos (x_{2}) + \cos (x_{1}) \sin (x_{2})$ and $\cos (x_{1} + x_{2}) = \cos (x_{1}) \cos (x_{2}) - \sin (x_{1}) \sin (x_{2})$ But I want to find a general formula for the sum of a finite number of angles for the Sine and the cosine but I didn't noticed any pattern. I suspect that it may have a recursive pattern.

Advadlippabrj 2022-01-27

Find the least value for $\sin x - \cos^{2} x - 1$
At first I found the first derivative to be $y^{'} = \cos x + 2 \sin x \cos x$
Critical point $0, - \frac{π}{6}$ (principal)
$y - \sin x + 2 (\cos 2 x)$
Then substitution of x by critical points I found minima. But my answer is incorrect.
Correct minimum value is $- \frac{9}{4}$

Vilharjevw7 2022-01-27

Prove that $4 \cos^{4} x - 2 \cos 2 x - \frac{1}{2} \cos 4 x$ is independent of x
I don't know how to proceed. $\cos 4 x = 2 \cos^{2} (2 x) - 1$

Gabriela Duarte 2022-01-27

Simplify $\sum_{k = 1}^{n} \tan (k) \tan (k - 1)$ by first proving $\tan (k) \tan (k - 1) = \frac{\tan (k) - \tan (k - 1)}{\tan (1)} - 1$

Ahmed Stewart 2022-01-27

Prove that $\cos (\frac{π z}{4}) - \sin (\frac{π z}{4}) = \prod_{n = 1}^{\infty} (1 + \frac{{(- 1)}^{n} z}{2 n - 1})$

derlingasmh 2022-01-26

Is there any simpler way to find $\sin 2 y$ from $\cos (x + y) = \frac{13}{}$ and $\cos (x - y) = \frac{15}{}$ ?

shangokm 2022-01-26

Express y through R and x.
How can I express y through R and x?
$y = R \pm \sqrt{R^{2} - {(R \arccos (\mp 1 \pm \frac{y}{R}) - x)}^{2}}$
It can be transformed into this :
$y = \pm R \cos (\frac{\sqrt{2 y R - y^{2}} + x}{R}) - R$

jelentetvq 2022-01-26

Find intersection of $| \frac{\cos (x)}{2} | and | \arctan (x) |$
I want to find the intersection for:
$| \frac{\cos (x)}{2} | = | \arctan (x) |, \forall x > 0$
my attempt: I tried to find the value of x as follows;
$\frac{\cos (x)}{2} = \frac{\cos (x)}{\sin (x)}$
Then
$\frac{1}{\sin (x)} = \frac{12}{}$
$\csc (x) = \frac{12}{}$
Then x does not have any answer

Stoockiltj5 2022-01-26

$\sqrt{2} x^{2} - \sqrt{3} x + k = 0$ with solutions $\sin θ, \cos θ$

Gerald Ritter 2022-01-26

What can we say about $a \cos^{2} θ + b \sin^{2} θ$ ?

Motalli21 2022-01-26

$\cos (2 π x), \cos (4 π x), \cos (8 π x), \cos (16 π x)$ , and $\cos (32 π x)$ are all nonpositive. What is the smallest positive x?

Duncan Reed 2022-01-26

If $u = \sqrt{a \cos^{2} x + b \sin^{2} x} + \sqrt{b \cos^{2} x + a \sin^{2} x}$ , find the maximum and minimum value of $u^{2}$ .

Swebaceacichegh 2022-01-26

Evaluating $\frac{{\cos 81}^{\circ}}{{\sin 3}^{\circ} {\sin 57}^{\circ} {\sin 63}^{\circ}}$

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.

Precalculus

Solve Precalculus Problems | Expert Help Available