Solving Trigonometry Equations and Problems with Examples and Tips

Verify the trig identity $\frac{\cot^{2} θ - 1}{\csc^{2} θ} = \csc θ - 1$

compagnia04 2022-01-16

If $z = \frac{{(\sqrt{3} + i)}^{n}}{{(\sqrt{3} - i)}^{m}}$ , find the relation between m and n such that z is a real number.

agreseza 2022-01-16

Solution of trig equation $\sin x + \cos x = \sqrt{2}$

deiteresfp 2022-01-16

Showing that $\frac{1 - \sin \frac{5 π}{18}}{\sqrt{3} \sin \frac{5 π}{18}} = \tan \frac{π}{18}$

Margie Marx 2022-01-16

$\sin 2 x - \sin^{2} x = 20 \sin x + 10 \cos x - 1$ with $90 \leq x \leq 180$

Vikolers6 2022-01-16

Find a limit $lim_{x \to 0} \int_{0}^{x} \frac{\cos (t^{3})}{t + x} d t$

Kaspaueru2 2022-01-16

Solving ${\frac{58}{\cot 36}}^{\circ} = \cos^{3} x$ without substituting the trig values for $36^{\circ}$

eiraszero11cu 2022-01-16

Simplifying $\frac{\sin x + \sin x \tan^{2} x}{\tan x}$ to $\sec x$

tearstreakdl 2022-01-16

$2^{\sin (x) + \cos (y)} = 1, 16^{\sin^{2} (x) + \cos^{2} (y)} = 4$ (system of equations)

obrozenecy6 2022-01-15

Find the minimum value of f
$f (x) = \frac{\tan (x + \frac{π}{6})}{\tan x}, x \in (0, \frac{π}{3})$
My approach is as follows. I tried to solve it by segregating it
$f (x) = \frac{1}{\sqrt{3} \tan x} + (\sqrt{3} + \frac{1}{\sqrt{3}}) \frac{1}{\sqrt{3} - \tan x}$
but f′(x) is getting more and more complicated

aramutselv 2022-01-15

For a minor class i am taking this year, I found the following integral in a problem set, and where i had no luck in evaluating it:
$\int \cos (2 \cot^{- 1} \sqrt{\frac{1 - x}{1 + x}}) d x$
I proceeded as follows:
->first, let $x = \cos (2 θ) \Rightarrow d x = - 2 \sin 2 θ d θ$ so the integral becomes:
$\int \cos (2 \cot^{- 1} \sqrt{\frac{1 - \cos (2 θ)}{1 + \cos (2 θ)}}) - 2 \sin 2 θ d θ = \int \cos (2 \cot^{- 1} \sqrt{\frac{\sin^{2} θ}{\cos^{2} θ}}) - 2 \sin 2 θ d θ$
$= \int \cos (2 \cot^{- 1} (\tan θ) - 2 \sin 2 θ d θ) = \int \cos \frac{2}{θ} - 2 \sin 2 θ d θ$
After this i am stuck. How do i proceed?

Juan Hewlett 2022-01-15

Finding $lim_{x \to a} \frac{1}{{(a^{2} - x^{2})}^{2}} (\frac{a^{2} + x^{2}}{a x} - 2 \sin (\frac{a π}{2}) \sin (\frac{x π}{2}))$
My attempt:
I considered the substitution $x = lim_{h \to 0} a \cos h$ and got the limit to be:
$lim_{h \to 0} \frac{1}{a^{4} h^{4}} (\frac{1 + \cos^{2} h}{\cos h} - 2 \cos^{2} (a h^{2} π))$

socorraob 2022-01-15

How to find the intervals on which f is concave up? $f (x) = \arctan (\sin x)$
$f (x) = \arctan (\sin x)$
$f^{'} (x) = \frac{\cos x}{1 + \sin^{2} x}$
$f^{″} (x) = - \frac{(\sin x) (2 + \cos^{2} x)}{{(1 + \sin^{2} x)}^{2}}$

Quentin Johnson 2022-01-15

Simplifying $\sec^{2} x - \tan^{2} x$
I am trying to solve this problem and I am stuck because $\sec^{2} x$ is a reciprocal of cosine and $\tan^{2} x$ is a function of $\frac{\sin (x)}{\cos (x)}$

hadejada7x 2022-01-15

Given $\cos (a) + \cos (b) = 1$ , prove that $1 - s^{2} - t^{2} - 3 s^{2} t^{2} = 0$ , where $s = \tan (\frac{a}{2})$ and $t = \tan (\frac{b}{2})$

Concepcion Hale 2022-01-15

For all real numbers x except x=0 and x=1 the function f(x) is defined by
$f (\frac{x}{x - 1}) = \frac{1}{x}$
Suppose $0 \leq t \leq \frac{π}{2}$ . What is the value of $f (\sec^{2} t)$ ?
I first set $\sec^{2} t = \frac{x}{x - 1}$ and then plugged it in to WA which told me that $x = \csc^{2} t$ so that would mean that $\frac{1}{x} = \frac{1}{\csc^{2} x}$ . However, that turned out to be wrong, so what should I do here?

Joan Thompson 2022-01-15

Using the identity $\tan θ = \tan (θ - 180^{\circ})$ to find values of $θ$ such that $\tan θ = {\tan 20}^{\circ}$

reproacht3 2022-01-15

Find maximum value of $(1 + \sin x) (1 + \cos x)$

Victor Wall 2022-01-15

Solve this equation for $x \in (0, \frac{π}{2})$
$\cos (2 x) \cos (x - \frac{π}{6}) = \sin (2 x) \sin (\frac{π}{6} - x)$
I gave a try using the $\cos (a - b)$ and $\sin (a - b)$ formulas, but it seems the problem complicated a little bit more.

Ikunupe6v 2022-01-15

Calculate $\frac{3 \sin α + 2 \tan α}{\cot α}$ if $\cos α = \frac{1}{3}$

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