Solving Trigonometry Equations and Problems with Examples and Tips

Find the value of $\log \frac{a + b i}{a - b i}$

Kinsley Moon 2022-01-24

Why is $\frac{\cos (α + β) + \cos (- α)}{\sin (α + β) + \sin (- α)}$ independent of $α$ ?

poveli1e 2022-01-24

Write $\cos^{2} (x)$ as linear combination of $x \mapsto \sin (x)$ and $x \mapsto \cos (x)$

Lainey Goodwin 2022-01-24

Consider the following equality:
$a_{1} \sin (x) + a_{2} \sin^{2} (x) + \dots + a_{n} \sin^{n} (x) = a_{1} \sin (x) + a_{2} \sin (2 x) + \dots + a_{n} \sin (n x)$
where $a_{1}, a_{2} \dots, a_{n} \in C, n \in N$
This equality clearly holds for n=1 and for any $a_{1} \in C$ . The trivial equality ( $a_{1} = a_{2} = \dots = a_{n} = 0$ ) also holds.
Do there exists any other solutions? If so, are there infinitely many of such solutions?
Do there exist other solutions if we constrain the domain of coefficients to Q, R or Z?

Mlejd5 2022-01-24

Find the number of roots of the equation, $x^{3} + x^{2} + 2 x + \sin x = 0$ in $[- 2 π, 2 π]$

Jessie Jenkins 2022-01-24

Reaching upon 9=1 while solving x for $3 \tan (x - 15^{\circ}) = \tan (x + 15^{\circ})$
Substituting $y = x + 45^{\circ}$ , we get
$3 \tan (y - 60^{\circ}) = \tan (y - 30^{\circ})$
$3 \frac{\tan y - \sqrt{3}}{1 + \sqrt{3} \tan y} = \frac{\tan y - \frac{1}{\sqrt{3}}}{1 + \frac{1}{\sqrt{3}} \cdot \tan y}$

$3 (\tan^{2} y - 3) = 3 \tan^{2} - 1$

9=1

The solution provided by the book $x = n π + \frac{π}{4}$ fits, so why did i get 9=1?

Rudy Koch 2022-01-24

If $2 (\sin a + \cos a) \sin b = 3 - \cos b$ , then find $3 \tan^{2} a + 4 \tan^{2} b$
I doubt there is a solution on the internet now, but they do have a number 35 as the final answer.
My tiny little progress:
Use simple simplification,
$3 \tan^{2} a + 4 \tan^{2} b = \frac{3}{\cos^{2} a} + \frac{4}{\cos^{2} b} - 7$
Intuitively, in order to get this, the given equation should be squared. But that is where I got stuck, because the term cosb seems hard to handle; it just seems different from other terms.

Berasiniz1 2022-01-24

Finding value of $\int_{0}^{\infty} \frac{x \sin x}{{(x^{2} + 1)}^{3}} d x$

Maximus George 2022-01-24

$x^{2} + y^{2} + x y = 1$ , then find the minimum of $x^{3} y + x y^{3} + 4$

Hailee Cline 2022-01-24

Volume of a solid $y = \cos (x)$ and y=0 for the interval $0 \leq x \leq \frac{π}{2}$
I used method of shells to get:
$2 π \int_{0}^{\frac{π}{2}} x \cos (x) d x$
And I got:
$2 π \int_{0}^{\frac{π}{2}} x \cos (x) d x = 2 π {[x \sin (x) + \cos (x)]}_{0}^{\frac{π}{2}}$
$= 2 π [\frac{π}{2} \sin (\frac{π}{2}) - \cos (0)]$
$= 2 π (\frac{π}{2} - 1)$
$2 π \int_{0}^{\frac{π}{2}} x \cos (x) d x = π^{2} - 2 π$
Then I am doing it by method of slicing and got this as an answer

Rylan Duncan 2022-01-24

Finding the infinite sum involving $\coth$ function using contour integration
I am looking to show: $\sum_{n = 1}^{\infty} \frac{\coth (n π)}{n^{3}} = \frac{7 π^{3}}{180}$

Kaydence Huff 2022-01-24

Different ways gives different results - solving $\tan 2 a = \sqrt{3}$
Case 1)
$\tan 2 a = \sqrt{3} = \tan (\frac{π}{3})$
$2 a = n π + \frac{π}{3}$
$a = \frac{n π}{2} + \frac{π}{6}, n \in Z$
Case 2)
$\frac{2 \tan a}{1 - \tan^{2} a} = \sqrt{3}$
Solving above equation gives $\tan a = \frac{1}{\sqrt{3}} or - \sqrt{3}$
Thus either
$a = n π + \frac{π}{6}, n \in Z$
Or
$a = m π - \frac{π}{3}, m \in Z$
Or either I did something wrong or both these results are same.

Alvin Pugh 2022-01-24

Show $\int_{0}^{2 π} \cos^{2} t d t = π$ using Pythagorean Theorem
The traditional way I always see this integral calculated is with the identity
$\cos^{2} t = \frac{1 + \cos (2 t)}{2}$
My alternative method uses $\cos^{2} t + \sin^{2} t = 1$ . Its

Connor Randall 2022-01-23

Evaluating $lim_{x \to 0} {(\sin x)}^{x}$

Autohelmvt 2022-01-23

If A+B+C=180, then $\frac{\sin 2 A + \sin 2 B + \sin 2 C}{\cos A + \cos B + \cos C - 1} = 8 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$
Then 2A+2B+2C=360
So
$\sin 2 C = - \sin (2 A + 2 B)$
Putting that in the equation
$2 \sin (A + B) \sin (A - B) - 2 \sin (A + B) \cos (A + B) \cos A + \cos B - \cos (A + B) + 1$
$2 \sin (A + B) [\sin (A - B) - \cos (A + B)] \cos A + \cos B - \cos (A + B) + 1$
I don’t know how to proceed. Please help me continue

Franco Franco 2022-01-23

Proving that $\sec \frac{π}{30} = \sqrt{2 - \sqrt{5} + \sqrt{15 - 6 \sqrt{5}}}$
which I instantly wanted to prove.
I know that I can "reduce" the problem to the evaluation of $\cos \frac{π}{15}$ , as the rest is easy with the use of the half-angle formula.
I know that cos obeys the nice relation
$\cos n x = T_{n} (\cos x)$
where
$T_{n} (x) = \frac{n}{2} \sum_{k = 0}^{| \frac{n}{2} |} \frac{{(- 1)}^{k}}{n - k} (\begin{matrix} n - k \\ k \end{matrix}) {(2 x)}^{n - 2 k}$
Thus, setting $t = \cos \frac{π}{15}$ ,
$T_{15} (t) = - 1$
The only thing left to do is solve for t. We can narrow down our search to the values 0<t<1.
I have never dealt with degree-15 polynomials before, so I was hoping one of you could help me out.

Selena Cowan 2022-01-23

what is x for $\tan^{2} 3 x = 2 \sin^{2} 3 x$

Alisha Pitts 2022-01-23

How to solve $\ln (\csc (z)) = \ln (2) + i z$ ?

Kinsley Moon 2022-01-23

How to simplify the ratio $\frac{1 - \cos 2 x + i \sin 2 x}{1 + \cos 2 x - i \sin 2 x}$

treslagosnv 2022-01-23

I have troubles with such limit
$lim_{(x, y) \to (0, 0)} \frac{x^{3}}{y^{4} + 2 \sin^{2} x}$
I approach on any line or curve I get limit equal to 0. I try polar coordinates - also nothing, still 0. I try to bound it somehow - I get infinity. However, wolfram tells that there is no limit for the function in (0,0)

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