Solving Trigonometry Equations and Problems with Examples and Tips

Evaluating $lim_{x \to 0} \frac{x - \tan^{- 1} x}{x \sin x}$

Stoockiltj5 2022-01-23

When do we have to change the limits?
$\int_{0}^{1} \sqrt{x - x^{2}} d x$
completing the square gives me:
$\int \sqrt{- x^{2} + x - \frac{1}{4} + \frac{1}{4}} d x$
$\int \sqrt{- (x - \frac{1}{2})^{2} + \frac{1}{4}} d x$
$\int \sqrt{- u^{2} + \frac{1}{4}} d u$
$\int \sqrt{\frac{1}{4} - u^{2}} d u$
Is trig sub the best way to go?
if $u = \frac{1}{2} \sin θ$ then $d u = \frac{1}{2} \cos θ d θ$
$= \int \sqrt{\frac{1}{4} - \frac{1}{4} \sin^{2} θ} \frac{1}{2} c o s θ d θ$
$\int \frac{1}{2} c o s θ \frac{1}{2} c o s θ d θ$
$\frac{1}{4} \int c o s^{2} θ d θ$
using half angle identity I eventually get:
$\frac{1}{4} \int \frac{1}{2} (1 + c o s (2 θ)) d θ$
$\frac{1}{8} \int (1 + c o s (2 θ) d θ$
$\frac{1}{8} (θ + \frac{1}{2} s i n 2 θ)$
Where do I go from here?

Octavio Miller 2022-01-23

Is ${tanh}^{- 1} (\sin 2 (x + d \frac{π}{4})) = \frac{1}{π} \ln (\cot^{2} x)$ ?
As you see the above graph that $\arctan h (\sin 2 (x + \frac{π}{4}))$ is similar to $\frac{1}{π} \ln (\cot^{2} x)$
Does this mean that by some minor alterations in the functions, they would be equal?
Why are both of functions so similar?

Franco Franco 2022-01-23

If $m^{2} \cos \frac{2 π}{15} \cos {\frac{4 π}{15}} \cos {\frac{8 π}{15}} \cos {\frac{14 π}{15}} = n^{2}$ then find the value of $\frac{m^{2} - n^{2}}{n^{2}}$

Keaton Good 2022-01-23

Proving $\int_{0}^{\infty} x^{n} e^{- t x} \frac{\sin x}{x} d x = \frac{\sin n θ}{{(1 + t^{2})}^{\frac{n}{2}}} (n - 1)!$ where $θ = \arcsin \frac{1}{\sqrt{1 + t^{2}}}$
Now I didnt

William Montgomery 2022-01-23

If $x = a {\cos t}^{3}, y = b {\sin t}^{3}$ then what is $\frac{d^{3} y}{{d x}^{3}}$ ?

Anika Osborne 2022-01-23

Let $θ \in (0, \frac{π}{4})$ and $t_{1} = (\tan θ)^{\tan θ}, t_{2} = (\tan θ)^{\cot θ}, t_{3} = (\cot θ)^{\tan θ}$ and $t_{4} = (\cot θ)^{\cot θ}$ , then show that $t_{4} > t_{3} > t_{1} > t_{2}$ .

iocasq4 2022-01-23

Find the smallest value of $f (x) = (\frac{19}{+} \frac{32}{\sin (x)}) (\frac{1}{32} + \frac{9}{\cos x})$ on the interval $(0, \frac{π}{2})$

Anika Klein 2022-01-23

If $\cos (n π) = {(- 1)}^{n}$ , then why is $- \frac{4}{n} \cos (n π) = \frac{4}{n} {(- 1)}^{n + 1}$ ?
I lectures has this written down in one of the solutions to an exercise however I'm not sure how he got $\frac{4}{n} {(- 1)}^{n + 1}$ ? I thought it would just be $- \frac{4}{n}$ . ${(- 1)}^{n}$ ? Am I missing something?

Alisha Pitts 2022-01-23

How do I prove $\tan 1 < \frac{π}{2}$ ?
Prove that the equation
$\sin x \sin (\sin x) = \frac{π}{2} \cos (\sin x)$
Let $t = \sin x, - 1 \leq t \leq 1$ . Then the expression above is equvalent to $t \sin t = \frac{π}{2} \cos t$ . As the function $f (t) = t \sin t - \frac{π}{2} \cos t$ is even, and t=0 is not a solution, I have to prove that f(t) has no positive roots (t>0). So, for the left side $0 < t \leq 1$ and $0 < \sin t \leq \sin 1$ , then $t \sin t \leq \sin 1$ . For the right side $\cos t \geq \cos 1$ , so $\frac{π}{2} \cos t \geq \frac{π}{2} \cos 1$ . The objective is to prove that $\sin 1 < \frac{π}{2} \cos 1$ , or, equivalently, $\tan 1 < \frac{π}{2}$

Jacquelyn Sanders 2022-01-23

If $z = \sqrt{3} + i = (a + i b) (c + i d)$ , then find $\tan^{- 1} \frac{b}{a} + \tan^{- 1} \frac{d}{c}$

Rudy Koch 2022-01-23

Solving $\sin 2 x \sin x + \cos^{2} x = \sin 5 x \sin 4 x + \cos^{2} 4 x$

estadsocm3 2022-01-23

Trigonometric Equation: $4 \sin θ = 3 \tan θ$
Solve this equation for $- 180^{\circ} \leq θ \leq 180^{\circ}$

Wendy Gutierrez 2022-01-23

How does $\frac{- e^{- x}}{\sqrt{e^{- 2 x} - 1}}$ rearrange?

Turnseeuw 2022-01-23

$\sin (\arccos (\frac{1}{\sqrt{3}})) = \frac{\sqrt{2}}{\sqrt{3}}$ proof this equation

bistandsq 2022-01-22

how to solve $\frac{\sin^{2} (θ) - 3 \cos^{2} (θ) + 1}{\sin^{2} (θ) - \cos^{2} (θ)} \equiv 2$ and similar equations

arrebolyt 2022-01-22

A parametric curve is described by
$x = \sin t, y = \sin (t + \frac{π}{6})$
Is there a way to show that
$y = \frac{\sqrt{3}}{2} x + \frac{\sqrt{1 - x^{2}}}{2}$
without using the property $\sin (a + b) = \sin a \cos b + \sin b \cos a$ ?

urlacainnxp 2022-01-22

let be $x \in R$
I want to show that if
$lim_{n \to \infty} \sin (n x) = 0$
then
$\sin (x) = 0$

2alr8w 2022-01-22

Let $f (x) = \tan^{- 1} x, x \in R$ , then the sequence ${f^{(n)} (0)}$ is unbounded. True or false?

Haialarmz6 2022-01-22

Find a solution to the equation
$\frac{1}{\sin^{2} (2 x)} + \tan (x) - \frac{1}{\tan (x)} = 2$
Assuming $\sin (x) \neq 0$ and $\cos (x) \neq 0$ , I simplified the above to
$4 \sin^{2} (x) - 1 - 4 \sin (x) \cos (x) = 0$
which is equivalent to
$\cos (2 x) + \sin (2 x) = \frac{1}{2}$
I do not know what to do next. I would be grateful for any hints.

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