Solving Trigonometry Problems: Basic, Advanced & Examples

High School
Geometry
Trigonometry

How to find first derivative of function $y = x \ln (x)$ by limit definition, that is using this formula
$y^{'} = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

oliviayychengwh 2022-01-02

How to prove this inequality in interval: $(0, \frac{π}{2}) 2 \cos x + \sec^{2} x - 3 \geq 0$

Kelly Nelson 2022-01-01

Can anyone see a way to simplify one of these expressions?
$\cos^{2} θ \sin ϕ + \sin^{2} θ \cos ϕ$
or
$\sin^{2} θ \sin ϕ - \cos^{2} θ \cos ϕ$
Ive

Kathy Williams 2022-01-01

what is the derivative of $\sec^{2} x$
I'm using the quotient rule and so I rewrite $\sec^{2} x as \frac{1}{\cos^{2} x}$
Then set $u = 1, u' = 0 then v = \cos^{2} x$ and $v' = - 2 \sin x \cos x$
I then get $\frac{2 \sin x \cos x}{\cos^{4} x}$
but when I simplify that I don't get $2 \sec^{2} x \tan x$ which is the correct answer?

Patricia Crane 2022-01-01

Solve for x which is in $[0, 2 π]$
$6 \cos x + 2 \sqrt{2} \sin x = \sqrt{22}$
I have solved the question by dividing both sides $\sqrt{44}$ , and got the answer that involves arcsin function. My question is:
Is it possible to solve it without any arc functions?

Mary Hammonds 2021-12-31

Find the value of x from $\sin x + \cos x = \sqrt{2} \sin (5 x)$
I used auxiliary argument method and converted into $\cos (\frac{π}{4} - x) = \cos (\frac{π}{2} - 5 x)$ (by introducing $\sqrt{2}$ as auxiliary argument and then using $\cos (\frac{π}{4}) \cos x + \sin (\frac{π}{4}) \sin x = \cos (\frac{π}{4} - x)$ and using $\sin 5 x = \cos (\frac{π}{2} - 5 x)$
But the answer isn't matching after using the formula for $\cos θ = \cos α$

zagonek34 2021-12-31

How to solve $\frac{1}{2 {\sin 50}^{\circ}} + 2 {\sin 10}^{\circ}$

Monique Slaughter 2021-12-31

If $\frac{1 - \sin x}{1 + \sin x} = 4$ , what is the value of $\tan \frac{x}{2}$ ?
1)-3
2)2
3)-2
4)3

Jason Yuhas 2021-12-31

Find the value of $\int_{- π}^{π} (4 \arctan (e^{x}) - π) d x$
Ive

Betsy Rhone 2021-12-31

Prove that $\frac{\sin (x)}{2} + \sin^{2} (\frac{x}{2}) \tan (\frac{x}{2}) \to \tan (\frac{x}{2})$

Pamela Meyer 2021-12-31

How do you evaluate $\tan (\frac{π}{6})$ ?

Gwendolyn Willett 2021-12-31

Find the value of $\cos^{- 1} (\sqrt{\frac{2 + \sqrt{3}}{4}})$
I am trying to solve:
$\sin^{- 1} \cot (\cos^{- 1} (\sqrt{\frac{2 + \sqrt{3}}{4}}) + \cos^{- 1} (\frac{\sqrt{12}}{4}) + \csc^{- 1} (\sqrt{2}))$
My solution is as follow:
$T = \sin^{- 1} \cot (\cos^{- 1} (\sqrt{\frac{2 + \sqrt{3}}{4}}) + \cos^{- 1} (\frac{\sqrt{12}}{4}) + \csc^{- 1} (\sqrt{2}))$
Since:
$\csc^{- 1} \sqrt{2} = \sin^{- 1} (\frac{1}{\sqrt{2}}) = \frac{π}{4}; \cos^{- 1} (\frac{\sqrt{12}}{4}) = \cos^{- 1} (\frac{\sqrt{3}}{2}) = \frac{π}{6}$
Then:
$T = \sin^{- 1} \cot (\cos^{- 1} (\sqrt{\frac{2 + \sqrt{3}}{4}}) + \frac{π}{4} + \frac{π}{6})$
I am not able to proceed further.

pierdoodsu 2021-12-31

Here are all the results I got
$\tan (A + B + C) = a - b$
And
$(1 + \tan^{2} A) (1 + \tan^{2} B) (1 + \tan^{2} C) = {(\frac{1}{\cos A \cos B \cos C})}^{2}$
And
$\cot A + \cot B + \cot C = 0$
How should I use these results?

Lorraine Harvey 2021-12-31

The Question
$\int \csc (x - \frac{π}{3}) \csc (x - \frac{π}{6}) d x$
What I Tried- I tried dividing both numerator and denominator by $\sin \frac{π}{6}$ but couldnt

Bobbie Comstock 2021-12-31

How do I evaluate the limit $lim_{x \to 0} \frac{\sin 3 x}{x \cos 2 x}$ ?
I'm having trouble doing this problem: the farthest I've gotten is just using a limit law for division and then moving the constant x in the denominator in front. I also thought about leaving $lim_{x \to 0} \frac{\sin 3 x}{1}$ and then somehow introducing 3x so it turns into a special limit.
Otherwise, I'm just stuck.

lugreget9 2021-12-31

I can not find a good way to solve this rather simple-looking equation. $\cos x + \cos \sqrt{2 x} = 2$
I can see that 0 is a solution, but is there a good way of solving it for all the potential solutions.

Brock Brown 2021-12-31

For any square matrix A, we can define sinA using the formal power series as follows:
$\sin A = \sum_{n = 0}^{\infty} \frac{{(- 1)}^{n}}{(2 n + 1)!} A^{2 n + 1}$
Prove or disprove: there exists a $2 \times 2$ real matrix A such that
$\sin A = [\begin{array}{cc} 1 & 2020 \\ 0 & 1 \end{array}]$

Ashley Bell 2021-12-31

Solving two trigonometric equations
How can I find x,y for:
$\sin x + \sin y = 0$
$\sin 2 x + \sin 2 y = 0$

Mary Reyes 2021-12-31

Find range of the function
$f (x) = 3 | \sin x | - 4 | \cos x |$
I tried to do by using the trigonometric identities
$\sin^{2} x = \frac{1 - \cos 2 x}{2}; \cos^{2} x = \frac{1 + \cos 2 x}{2}$
So
$f (x) = 3 \sqrt{\frac{1 - \cos 2 x}{2}} - 4 \sqrt{\frac{1 + \cos 2 x}{2}}$
but don't know how to proceed from here

interdicoxd 2021-12-31

Find the value of $\tan (\frac{π}{3})$

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Trigonometry

Trigonometry Problems - Basics, Advanced & Examples