Solving Trigonometry Equations and Problems with Examples and Tips

Suppose we are given a function
$g (x) = \sum_{n = 1} \frac{\sin (n x)}{10^{n} \sin (x)}, x \neq k π, k \in Z$

Nathanael Hansen 2022-03-31

I have to prove that
$\prod_{k = 1}^{n - 1} (1 - cis (\frac{2 k π}{n})) = n$

svrstanojpkqx 2022-03-31

Summation of $\sum_{r = 1}^{n} \frac{\cos (2 r x)}{\sin ((2 r + 1) x \sin ((2 r - 1) x}}$
My Try:
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x) \sin ((2 r + 1) x - (2 r - 1) x)}{\sin 2 x : \sin ((2 r + 1) x \sin ((2 r - 1) x}}$
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x) (\sin ((2 r + 1) x \cos (2 r - 1) x - \cos (2 r - 1) x) \sin (2 r + 1) x)}{\sin 2 x \sin ((2 r + 1) x \sin ((2 r - 1) x)}}$
$S = \sum_{r = 1}^{n} \frac{\cos (2 r x)}{\sin 2 x} (\cot (2 r - 1) x - \cot (2 r + 1) x)$

tambustqaze 2022-03-31

Sum of $1 - {\cos θ}_{i}$ is bounded
For each $i = 1, \dots, N$ let $θ_{i} \geq 0$ be angles such that $\sum_{i = 1}^{N} θ_{i} \leq π$ . Prove that $\sum_{i = 1}^{N} (1 - {\cos θ}_{i}) \leq 2$

Kendall Daniels 2022-03-31

I have $\frac{e^{y i} - e^{- y i}}{e^{y i} + e^{- y i}}$ , and I know that this equals
$\frac{\frac{1}{2} (e^{y i} - e^{- y i})}{\frac{1}{2} (e^{y i} + e^{- y i})} = \frac{sinh {y i}}{cosh {y i}} = tanh {y i}$
However, this is also supposed to equal $i \tan y$ . I'm not sure how to get to this result.
I tried doing
$\frac{e^{y i} - e^{- y i}}{e^{y i} + e^{- y i}} = \frac{\frac{e^{2 y i} - 1}{e^{y i}}}{\frac{e^{2 y i} + 1}{e^{y i}}} = \frac{e^{2 y i} - 1}{e^{2 y i} + 1}$

burubukuamaw 2022-03-31

I have a problem to find answer for this limit.
$lim_{x \to 0} \frac{1 - \cos^{\frac{1}{3}} x}{1 - \cos (x^{\frac{1}{3}})}$

justbethlflql 2022-03-31

Studying convergence of $\int_{a}^{+ \infty} \frac{\sin (x)}{x^{n}}$ for $a, n \in R^{+}$ ?
There is two cases a=0 and $a \neq 0$ . This was my attempt. Case a=0:
$\int_{0}^{+ \infty} \frac{\sin (x)}{x^{n}} = \int_{0}^{1} \frac{\sin (x)}{x^{n}} + \int_{1}^{+ \infty} \frac{\sin (x)}{x^{n}}$
But
$\int_{0}^{1} \frac{\sin (x)}{x^{n}} = \int_{0}^{1} \frac{x}{x^{n}} = \int_{0}^{1} \frac{1}{x^{n - 1}}$
converges for n<2 and diverges for $n \geq 2$ . And
$\int_{1}^{+ \infty} \frac{- 1}{x^{n}} < \int_{1}^{+ \infty} \frac{\sin (x)}{x^{n}} < \int_{1}^{+ \infty} \frac{1}{x^{n}}$
converges for n>1 and diverges for $n \leq 1$ . That means that $\int_{0}^{+ \infty} \frac{\sin (x)}{x^{n}}$ converges for $n \in [1, 2]$ and diverges for $n \in R - [1, 2]$ .

abitinomaq1 2022-03-31

I can't find a way to solve this:
$\int_{\frac{π}{2}}^{π} \sqrt{8 \sin^{2} (t) \cos^{2} (t)} d t$

xxgetthisar08 2022-03-31

I cannot find the one-sided limit
$lim_{x \to \frac{π}{2} + 0} \frac{\sqrt{1 + \cos 2 x}}{\sqrt{π} - \sqrt{2 x}}$

ezpimpin6988ok1n 2022-03-31

Solving the system: $11 {\cos x}_{1} + 12 {\cos x}_{2} = 0$ and $11 {\sin x}_{1} + 12 {\sin x}_{2} = 17$

Pablo Dennis 2022-03-31

Solving this trigonometric equation:
$\sqrt{3} \cos x - 3 \sin x = 4 \sin 2 x \cos 3 x$

Oliver Carson 2022-03-31

Solving the equation $\tan {x} + \tan \frac{x}{4} = 2$
1st attempt: I use the following identity $\tan 2 x = \frac{2 \tan x}{1 - \tan^{2} x}$ . Then get the following equation
$u^{5} - 2 u^{4} - 8 u^{3} + 12 u^{2} + 3 u + 2 = 0$
where $u = \tan \frac{x}{4}$ . But i can't find any root for this equation.
2nd attempt:
$\tan {4 x} + \tan x = 2 \to \sin {4 x} \cos {x} + \sin {x} \cos {4 x} = 2 \cos {4 x} \cos {x} \to \sin {5 x} = 2 \cos {4 x} \cos {x}$
$\sin {5 x} = 2 \cos {4 x} \cos {x} \to \frac{1}{2} \sin {5 x} = \cos {4 x} \cos {x} \to \cos ({2 k π \pm \frac{π}{3}}) \sin {5 x} = \cos {4 x} \cos {x}$
I can't continue from here.

alparcero97oy 2022-03-31

I am trying to completely simplify the derivative of the following function:
$f (x) = \sqrt{16 - x^{2}} - 4 \cos^{- 1} (\frac{x}{4})$

Anahi Solomon 2022-03-31

How would you solve:
$2 \csc^{2} x = 3 \cot^{2} x - 1$

acidizihvzs 2022-03-31

How would you evaluate the integral
$\int_{0}^{π} (\frac{π}{2} - x) \sin (\frac{3 x}{2}) \csc (\frac{x}{2}) d x$

metalskaashw 2022-03-31

Solving $\tan (x + 1) - \tan (x) = 0$

Zoie Phillips 2022-03-31

How to solve the equation
$\tan^{2} (2 x) + 2 \tan (2 x) \cdot \tan (3 x) - 1 = 0$

Nathen Peterson 2022-03-31

How to solve the equation
$\cos^{2} (x) + \cos^{2} (2 x) + \cos^{2} (3 x) = 1$

Coradossi7xod 2022-03-31

$3 p \sin x - (p + \sin x) (p^{2} - p \sin x + \sin^{2} x) = 1$ has a solution for x. Then number of integral solutions of p are ?

ikramkeyslo4s 2022-03-31

$4 {\sin 72}^{\circ} {\sin 36}^{\circ} = \sqrt{5}$

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