Trigonometric Equations and Indentities: Real-World Examples

High School
Geometry
Trigonometry
Trigonometric equations and identities

Recent questions in Trigonometric equations and identities

Adela Brown 2022-01-02

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: $(0, \frac{π}{2}) 2 \cos x + \sec^{2} x - 3 \geq 0$

Dowqueuestbew1j 2022-01-02

f: $R^{+} \cup {0} \to R$
$f (x) = \arcsin (\cos \sqrt{x}) + \arccos (\sin \sqrt{x})$ then what is the derivative of function f?
A) $\frac{- 1}{\sqrt{x}}$
B) $\frac{1}{\sqrt{x}}$
c) $\frac{1}{2 \sqrt{x}}$

pierdoodsu 2022-01-02

Show ${(\sin (θ) - \cos (2 θ))}^{2} = 1 + \sin (θ) - \sin (3 θ) - \frac{12}{\cos (2 θ)} + \frac{12}{\cos (4 θ)}$

Shirley Thompson 2022-01-02

Find the extrema of $(1 + \sin x) (1 + \cos x)$ without using calculus.

prsategazd 2022-01-02

I want to solve this integral :
$\int_{0}^{\frac{π}{2}} \cot x \ln (\sec x) d x$
I tried the following substitution : $\ln (\sec x) = t$ which means $d t = \tan x d x$
$I = \int_{0}^{\infty} \frac{\cot x}{\tan x} t d t = \int_{0}^{\infty} \frac{t}{\tan^{2} x} d t$
I'm really disturbed by the $\tan^{2} x$ , I tried also to substitute $\sec x = t$ but it's not helpful either. Any helpful approach to solve this problem ?

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?

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

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?

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?

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.

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

Ashley Bell 2021-12-31

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

Betsy Rhone 2021-12-31

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

Jason Yuhas 2021-12-31

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

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

zagonek34 2021-12-31

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

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 $\cos θ = \cos α$

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.

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.

The majority of students do not like Trigonometry because they find all the questions and equations overly complex. Do not let it frighten you because we have lots of answers that will help you achieve solutions by turning to already existing tasks. We have practical works on all the twelve trigonometric identities and equations with 45 formulas that you can apply in your work. Since you can take a look at the resolved assignments, you will become a trigonometric identities and equations solver that can start with the formulas and continue with the graphics, analytical calculation, and related word problems.

Trigonometry

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