Solving Trigonometry Equations and Problems with Examples and Tips

$\sin (x) - x \cos (x) - \frac{1}{2} x^{2} = 0$
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?

Courtney Griffin 2022-01-26

Prove that $\cos^{4} (θ) - \sin^{4} (θ) = \cos (2 θ)$
So, here are my steps so far:
${(\cos^{2} θ)}^{2} - {(\sin^{2} θ)}^{2} = \cos (2 θ)$
$(\cos^{2} θ + \sin^{2} θ) (\cos^{2} θ - \sin^{2} θ) = \cos (2 θ)$
$(\cos^{2} θ + \sin^{2} θ) (\cos θ + \sin θ) (\cos θ - \sin θ) = \cos (2 θ)$

Gwendolyn Giles 2022-01-26

Can $\frac{\csc α + \cos α}{\cos α - \tan α - \sec α}$ be simplified?

Adrian Cervantes 2022-01-26

Eliminate $θ$ from from
$\frac{x - k \sin θ \cot α}{k \cos θ} = \frac{y - k \cos θ \tan α}{k \sin θ} = \frac{z}{c}$

William Montgomery 2022-01-26

I want to find the local minima of this equation
$\sin^{2} (\frac{33}{x} π) + \sin^{2} (x π) = y$

ardeinduie 2022-01-26

Calculate $\int_{- 4 π}^{4 π} \frac{{(\sin x)}^{2} - {(\sin x)}^{4}}{1 - {(\sin x)}^{4}} d x$

veceriraby 2022-01-26

Proving that $(\begin{matrix} n + 3 \\ 3 \end{matrix}) = \frac{n + 2}{2} \sum_{k = 1}^{n + 1} \csc^{2} \frac{k π}{n + 2}$

derlingasmh 2022-01-26

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

Haialarmz6 2022-01-26

Proving $\tan^{2} (\frac{π}{4} + x) = \frac{1 + \sin (2 x)}{1 - \sin (2 x)}$

Maximus George 2022-01-26

Showing that $\frac{x^{2} + 2 x \cos 2 α + 1}{x^{2} + 2 x \cos 2 β + 1}$ lies between $\frac{\cos^{2} α}{\cos^{2} β}$ and $\frac{\sin^{2} α}{\sin^{2} β}$

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

The majority of Trigonometry Math problems are quite easy to solve when you have the list of answers that help you see the logic and reasoning. Looking at various examples will help you to see how certain questions are related to provided solutions.

Remember that Trigonometry requires patience and analysis. By following your analysis skills, you will help yourself find the answers. If something remains unclear, start with your exploration of examples again and go step-by-step.

Precalculus

Practice Exercises for Trigonometry