Solving Trigonometry Equations and Problems with Examples and Tips

How to solve sum of $\cos (k x)$ for the case $\cos (x) = 1$
I have the solution for $\sum_{{k = 1}}^{n} \cos (k x)$ :
$\sum_{k = 1}^{n} \cos (k x) = R (\sum_{k = 1}^{n} e^{i k x})$
$= R (e^{i x} \frac{e^{i n x} - 1}{e^{i x} - 1}) = R (\frac{e^{i x} e^{\frac{i n x}{2}}}{e^{\frac{i x}{2}}} \frac{e^{\frac{i n x}{2}} - e^{\frac{- i n x}{2}}}{e^{\frac{i x}{2}} - e^{\frac{- i x}{2}}}) = R (e^{\frac{i (n + 1) x}{2}} \frac{\sin \frac{n x}{2}}{\sin \frac{x}{2}}) = \frac{\sin \frac{n x}{2}}{\sin \frac{x}{2}} \cos (\frac{(n + 1) x}{2})$
but if $e^{i x} = 1$ i am dividing trough 0 in line 2. So i know that in this case $e^{i x} \neq 1$ . So my question is: What is the $\sum_{k = 1}^{n} \cos (k x) if e^{i x} = 1$

Daniell Phillips 2022-01-17

Prove trigonometric an identity
$\frac{2 \cos α - 1}{\sqrt{3} - 2 \sin α} = \tan (\frac{α}{2} + \frac{π}{6})$

keche0b 2022-01-16

Integrate from 0 to $2 π$ with respect to $θ$ the following ${(\sin θ + \cos θ)}^{n}$

Ben Shaver 2022-01-16

Evaluate $\int_{0}^{\frac{π}{2}} \frac{1}{a \sin^{2} x + b \cos^{2} x} d x$

Danelle Albright 2022-01-16

Simplifying $\sin h (2 \arctan h (e^{H x}))$

Dowqueuestbew1j 2022-01-16

Questions about proving $lim_{x \to 0} \frac{\sin x}{x} = 1$

hionormf 2022-01-16

Calculation of $\int_{0}^{\frac{π}{4}} {(\tan (x))}^{n} d x$

Irrerbthist6n 2022-01-16

I want to solve this equation
$\arccos (x) + \arcsin (x^{2} - x + 1) = \frac{π}{2}$
I write this: for all $x \in [- 1, 1]$
$\arcsin (x^{2} - x + 1) = \frac{π}{2} - \arccos (x)$ then $x^{2} - x + 1 = \sin (\frac{π}{2} - \arccos (x)) = \cos (\arccos (x)) = x$
$x^{2} - x + 1 = x \Rightarrow x^{2} - 2 x + 1 = 0 \Rightarrow {(x - 1)}^{2} = 0 \Rightarrow x = 1$
Is it true ?

Terrie Lang 2022-01-16

If $\cos 17 x = f (\cos x)$ , then show that $\sin 17 x = f (\sin x)$

Harold Kessler 2022-01-16

Analytically solving $\frac{1}{\sin 2 x} + \frac{1}{\sin 3 x} = \frac{1}{\sin x}$

feminizmiki 2022-01-16

Evalute $\frac{\tan α - \cot α}{\sin^{4} α - \cos^{4} α}$ if $α$ is an acute angle and $\tan α = 2$

William Boggs 2022-01-16

Showing $\frac{d θ}{d \tan θ} = \frac{1}{1 + \tan^{2} θ}$
I suppose that
$\frac{d θ}{d \tan θ} = \frac{d \arctan x}{d x} = \frac{1}{1 + x^{2}} = \frac{1}{1 + \tan^{2} θ}$
So is
$\frac{d θ}{d \tan θ} = \frac{1}{1 + \tan^{2} θ}$
correct? And
$\frac{d θ}{d \tan \frac{θ}{2}} = \frac{2}{1 + \tan^{2} \frac{θ}{2}}$ ?

pogonofor9z 2022-01-16

Solutions of $e^{x} - 1 - k \cdot \arctan x = 0$ ,
consider $h (x) = e^{x} - 1 - k \cdot \arctan x$ then find on which condition on k there will be two solutions for h(x)=0 (k is real).

David Young 2022-01-16

The derivative of $f (x) = \frac{3 \sin x}{2 + \cos x}$
My solution:
The background
$3 \frac{d}{d x} (\frac{\sin x}{2 + \cos (x)})$
$= 3 \frac{\frac{d}{d x} (\sin (x)) (2 + \cos (x)) - \frac{d}{d x} (2 + \cos (x)) \sin (x)}{{(2 + \cos (x))}^{2}}$
$= 3 \frac{\cos (x) (2 + \cos (x)) - (- \sin (x)) \sin x}{{(2 + \cos (x))}^{2}}$
$= \frac{3 + 6 \cos (x)}{{(2 + \cos (x))}^{2}}$
However, I plugged $f (x) = \frac{3 \sin x}{2 + \cos x}$ into the derivative calculator in wolfram alpha and received the following calculation:
$\frac{3 (\sin^{2} (x) + \cos^{2} (x) + 2 \cos (x))}{{(2 + \cos (x))}^{2}}$
Is my solution incorrect and the one from wolfram alpha correct?

Lennie Davis 2022-01-16

Simplify $\frac{4 \cos^{2} (2 x) - 4 \cos^{2} (x) + 3 \sin^{2} (x)}{4 \sin^{2} (x) - \sin^{2} (2 x)}$

Francisca Rodden 2022-01-16

Prove that $\tan^{- 1} \frac{\sqrt{1 + x^{2}} + \sqrt{1 - x^{2}}}{\sqrt{1 + x^{2}} - \sqrt{1 - x^{2}}} = \frac{π}{4} + \frac{1}{2} \cos^{- 1} x^{2}$ ?
Let the above expression be equal to $ϕ$
$\frac{\tan ϕ + 1}{\tan ϕ - 1} = \sqrt{\frac{1 + x^{2}}{1 - x^{2}}}$
$\frac{1 + \tan^{2} ϕ + 2 \tan ϕ}{1 + \tan^{2} ϕ - 2 \tan ϕ} = \frac{1 + x^{2}}{1 - x^{2}}$
$\frac{1 + \tan^{2} ϕ}{2 \tan ϕ} = \frac{1}{x^{2}}$
$\sin 2 ϕ = x^{2}$
$ϕ = \frac{π}{4} - \frac{1}{2} \cos^{- 1} x^{2}$
Where am I going wrong?

Gwendolyn Willett 2022-01-16

Solving $\cos (z) = \frac{52}{}$
and I'm trying to solve for z but I keep going in circles. I know $\cos z = \frac{52}{=} \frac{12}{e^{i z} + e^{- i z}}$ so then $e^{i z} + e^{- i z} = 5$ but then I'm stuck

Mary Reyes 2022-01-16

Why is $\arctan (\tan (\frac{25 π}{4})) = \frac{π}{4}$ , and how can I get from the expression on the left to the one on the right?

Joseph Krupa 2022-01-16

I need some help answering this question:
$f (x) = \frac{cosh (x)}{sinh (x)} - \frac{1}{x}$
find the limit of f(x) as x tends to 0 by writing f(x) as a quotient of two powers series.
I have so far:
$\frac{(x (1 + \frac{x^{2}}{2!} + \frac{x^{4}}{4!} + \dots)) - (x + \frac{x^{3}}{3!} + \dots)}{x (x + \frac{x^{3}}{3!} + \dots)} = \frac{(x + \frac{x^{3}}{2!} + \frac{x^{5}}{4!} + \dots) - (x + \frac{x^{3}}{3!} + \dots)}{(x^{2} + \frac{x^{4}}{3!} + \dots)}$
but I don't know how to reduce this further.

veksetz 2022-01-16

Evaluate the following in terms of $α = lim_{x \to 0} \frac{\sin x}{x}$ :
$lim_{x \to 0} \frac{\tan^{2} (x) + 2 x}{x + x^{2}}$

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