Solving Trigonometry Problems: Basic, Advanced & Examples

High School
Geometry
Trigonometry

Hint to prove $\sin^{4} x + \cos^{4} x = \frac{3 + \cos (4 x)}{4}$

prsategazd 2021-12-30

I need to solve this limit:
$lim_{x \to 0} {\cos x}^{\frac{1}{\sin x}}$

lugreget9 2021-12-30

$lim_{x \to \frac{π}{4}} \frac{(\csc^{2} x - 2 \tan^{2} x)}{(\cot x - 1)}$ without L'Hôpitals Rule.

Roger Smith 2021-12-30

$lim_{x \to 0} \frac{\sin h - \sin x}{x (\cos h (x) - \cos (x))}$

Maria Huey 2021-12-30

Does the following limit exist?
$lim_{x \to 0} x \sin (\frac{1}{x})$
If yes, then I have to find it; if no, then I need to give reason why

abreviatsjw 2021-12-30

We have:
${\begin{matrix} 2 x = y \tan θ + \sin θ \\ 2 y = x \cot θ + \cos θ \end{matrix}$
And want to prove $x^{2} + y^{2} = 1$
My works:
I multiplied first equation by $\cos θ$ and second one by $\sin θ$ and get:
${\begin{matrix} 2 x \cos θ = y \sin θ + \sin θ \cos θ \\ 2 y \sin θ = x \cos θ + \sin θ \cos θ \end{matrix}$
By extracting $\sin θ \cos θ$ we get:
$2 x \cos θ - y \sin θ = 2 y \sin θ - x \cos θ$
$x \cos θ = y \sin θ$

Daniell Phillips 2021-12-30

If $\cos x \cdot \cos 2 x = \frac{14}{,} x \in [0, 90^{\circ})$ , then what is the solution of the equation?
I attempted to solve this question as follows:
$\cos 2 x = \cos^{2} x - \sin^{2} x$
$\Rightarrow \cos x (\cos^{2} x - \sin^{2} x) = \frac{1}{4}$
$\cos^{3} x - \sin x (\cos x \sin x)$
And I got stuck here, I did not know how to continue.

Osvaldo Apodaca 2021-12-30

How can I evaluate $lim_{v \to \frac{π}{3}} \frac{1 - 2 \cos v}{\sin (v - \frac{π}{3})}$ without using LHospitals rule?

kramtus51 2021-12-30

Prove that ${\tan 70}^{\circ} = {\tan 20}^{\circ} + 2 {\tan 50}^{\circ}$
My approach:
$L H S = {\tan 70}^{\circ} = \frac{1}{{\cot 70}^{\circ}} = \frac{1}{{\tan 20}^{\circ}}$
$R H S = {\tan 20}^{\circ} + 2 {\tan 50}^{\circ}$
$= {\tan 20}^{\circ} + 2 {\tan (20 + 30)}^{\circ}$
$= {\tan 20}^{\circ} + \frac{2 ({\tan 20}^{\circ} + \frac{1}{\sqrt{3}})}{1 - \frac{{\tan 20}^{\circ}}{\sqrt{3}}}$
$= \frac{2 + 3 \sqrt{3} {\tan 20}^{\circ} - \tan^{2} 20^{\circ}}{\sqrt{3} - {\tan 20}^{\circ}}$
Why are the two sides not equal despite being expressed in the same terms?

Mabel Breault 2021-12-30

A plane flies directly over a radar station while traveling at $500 \frac{m i}{h}$ passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 2 mi away from the station.
Finish solving the problem

Alfred Martin 2021-12-30

find integral $\int_{0}^{\frac{π}{2}} \frac{\sin (40 x)}{\sin 5 x} d x$
My attempt -writing
$\sin 40 x = \sin 40 x - \sin 30 x + \sin 30 x - \sin 20 x + \sin 20 x - \sin 10 x + \sin 10 x$

petrusrexcs 2021-12-30

Find the value of ${\cos 105}^{\circ} + {\sin 75}^{\circ}$
We can write the given trig expression as
$\cos (180^{\circ} - 75^{\circ}) + {\sin 75}^{\circ} = - {\cos 75}^{\circ} + {\sin 75}^{\circ}$
$= {\sin 75}^{\circ} - {\cos 75}^{\circ}$

Arthur Pratt 2021-12-30

Proving $\frac{2 \sin x + \sin 2 x}{2 \sin x - \sin 2 x} = \csc^{2} x + 2 \csc x \cot x + \cot^{2} x$
Proving right hand side to left hand side:
$\csc^{2} x + 2 \csc x \cot x + \cot^{2} x = \frac{1}{\sin^{2} x} + \frac{2 \cos x}{\sin^{2} x} + \frac{\cos^{2} x}{\sin^{2} x}$ (1)
$= \frac{\cos^{2} x + 2 \cos x + 1}{\sin^{2} x}$ (2)
$= \frac{1 - \sin^{2} x + 1 + \frac{\sin 2 x}{\sin x}}{\sin^{2} x}$ (3)
$= \frac{\frac{2 \sin x - \sin^{3} x + \sin 2 x}{\sin x}}{\sin^{2} x}$ (4)
$= \frac{2 \sin x - \sin^{3} x + \sin 2 x}{\sin^{3} x}$ (5)
I could not prove further to the left hand side from here.

Tara Alvarado 2021-12-30

How do I solve the following:
$\cos (12 x) = 5 \sin (3 x) + 9 \tan 2 (x) + \cot 2 (x) f or x \in (0, 360)$

Alan Smith 2021-12-30

How do I solve this?
$\cos 3 x = \cos^{2} x - 3 \sin^{2} x$

Shelia Lawrence 2021-12-30

I need to prove trigonometric equation
$\tan 2 x = \frac{2 \sin x \cdot \cos x}{\cos^{2} x - \sin^{2} x}$

Mabel Breault 2021-12-30

I want to solve for x
$2^{\sin^{4} x - \cos^{2} x} - 2^{\cos^{4} x - \sin^{2} x} = \cos 2 x$

rheisf 2021-12-29

What is the derivative of $x \sin x$

Kelly Nelson 2021-12-29

Find the general solution for given trigonometric equation
$\sin^{2} x \cos^{2} x + \sin x \cos x - 1 = 0$
The options are given in the form of $\tan^{- 1}$ , so I tried to convert the equation completely in tan but was unable to do so.
I also tried using the identity of $\sin 2 x$ , through which I got
$\sin^{2} 2 x + 2 \sin 2 x - 4 = 0$
I have got no idea how to proceed further.

Victor Wall 2021-12-29

Evaluate $\tan \frac{3 π}{2}$

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Trigonometry

Trigonometry Problems - Basics, Advanced & Examples