Solving Trigonometry Problems: Basic, Advanced & Examples

High School
Geometry
Trigonometry

Double Angle Formulas: Finding $\tan 2 θ$
I am trying to find $\tan 2 θ$ where $\sin θ = \frac{5}{13}$ and $θ$ is in Quadrant One.
According to my textbook, $\tan 2 θ = \frac{120}{119}$ , but I get $\frac{- 10}{13}$ instead.
The Identity I am using:
$\tan 2 θ = \frac{2 \tan θ}{1 - \tan^{2} θ}$

siutulysr5 2022-01-28

Find $\sin x and \cos x$ knowing $\tan x$
I know $\tan x = - 2 \sqrt{2}$ . How to find $\sin x$ and $\cos x$ if $x \in [- \frac{π}{2}, 0]$ ? They probably would be $- \frac{2 \sqrt{2}}{3} and \frac{13}{}$ respectively but I don't know how to prove it.

treetopssan 2022-01-27

Find the set of values, which are taken by $\tan z$
I was trying to write tangents as follows:
$\tan (z) = - i \frac{e^{i z} - e^{- i z}}{e^{i z} + e^{- i z}}$
and then
z=a+bi
, which led me to
$\tan z = - i \frac{\cos a (e^{- b} - e^{b}) + i \sin a (e^{- b} + e^{b})}{\cos a (e^{- b} + e^{b}) + i \sin a (e^{- b} - e^{b})}$

Kaydence Huff 2022-01-27

Proving $\frac{\csc x + \cot x}{\tan x + \sin x} = \cot x \csc x$

Natasha Gill 2022-01-27

How To Solve a Trigonometric Differential Equation
Salutations, I have been trying to approach an ODE with trigonometric functions that I found interesting:
$y^{'} + x \sin (2 y) = x e^{- x^{2}} \cos^{2} (y)$
I tried to find a result with wolfram web page (free version) and I got this one:
$y = \arctan (\frac{1}{2} e^{- x^{2}} (c + x^{2}))$
I have tried to approach this exercise by substitution of variables, also separable variables and I have not had luck by power series, and I do not know if methods like those of Ricatti and Bernoulli are appropriate for this case.

poveli1e 2022-01-27

I calculated ${\sin 75}^{\circ}$ as $\frac{1}{2 \sqrt{2}} + \frac{\sqrt{3}}{2 \sqrt{2}}$ , but the answer is $\frac{\sqrt{2} + \sqrt{6}}{4}$ . What went wrong?
I calculated the exact value of ${\sin 75}^{\circ}$ as follows:
${\sin 75}^{\circ} = \sin (30^{\circ} + 45^{\circ})$
$= \sin 30^{\circ} \cos 45^{\circ} + \cos 30^{\circ} \sin 45^{\circ}$
$= \frac{1}{2} \cdot \frac{1}{\sqrt{2}} + \frac{\sqrt{3}}{2} \cdot \frac{1}{\sqrt{2}}$
$= \frac{1}{2 \sqrt{2}} + \frac{\sqrt{3}}{2 \sqrt{2}}$
The actual answer is
$\frac{\sqrt{2} + \sqrt{6}}{4}$

kejpsy 2022-01-26

Find coordinate points where the tangent line is horizontal for $f (x) = - \sin (8 x) + 6 \cos (4 x) - 8 x$

Alvin Pugh 2022-01-26

Find $\sin \frac{θ}{2}$ when $\sin θ = \frac{3}{5}$ , and $0^{\circ} < θ < 90^{\circ}$
The answer I'm currently getting is $\sqrt{\frac{1}{10}}$ but the answer must be either $\frac{\sqrt{10}}{10}, \frac{3 \sqrt{10}}{10} or \frac{1}{3}$
My Process:
since $\sin \frac{θ}{2} = \pm \sqrt{\frac{1 - \cos θ}{2}}$
$\sqrt{\frac{1 - \frac{4}{5}}{2}} \cdot \sqrt{\frac{5}{5}} = \sqrt{\frac{5 - 4}{10}} = \sqrt{\frac{1}{10}}$

Seamus Kent 2022-01-26

Proving $\int_{0}^{2 π} (\cos (t)^{2 n}) = (\binom{2 n}{n}) \frac{2 π}{2^{2 n}}$ using the following result $\int_{γ} (z + \frac{1}{z})^{2 n} \frac{d z}{z} = (\binom{2 n}{n}) 2 π i$ Prove $\int_{0}^{2 π} (\cos (t)^{2 n}) = (\binom{2 n}{n}) \frac{2 π}{2^{2 n}}$

taoigas 2022-01-26

If the equation $\sin^{2} x - a \sin x + b = 0$ has only one solution in $(0, π)$ , then what is the range of b?

taoigas 2022-01-26

How do I prove $\frac{\sin x + \csc y}{\sin y + \csc x} = \csc y \sin x$ ?

Tessa Leach 2022-01-25

Find modulus and argument of $ω = \frac{\sin (P + Q) + i (1 - \cos (P + Q))}{(\cos P + \cos Q) + i (\sin P - \sin Q)}$

pozicijombx 2022-01-25

Given that $\tan \frac{x}{2} = \frac{1 - \cos (x)}{\sin (x)}$ , deduce that $\tan \frac{π}{12} = 2 - \sqrt{3}$
I know $\tan \frac{x}{2} = \frac{1 - \cos (x)}{\sin (x)}$ is true and I can prove it by squaring and taking a square root of the right side then I multiply by $\frac{0.5}{0.5}$ . And I will use
$\cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}}$
and
$\sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}}$
But I do not understand the part of deducing.

Ahmed Stewart 2022-01-25

If $\sin α + \sin β = a and \cos α - \cos β = b$ , then what is $\tan (\frac{α - β}{2})$ ?
If I square both the equations
$2 + 2 \sin (α - β) = a^{2} + b^{2}$
$\sin (α - β) = \frac{a^{2} + b^{2} - 2}{2}$
Since $\sin 2 θ = \frac{2 \tan θ}{1 + \tan^{2} θ}$ , then
$\sin (α - β) = \frac{2 \tan \frac{α - β}{2}}{1 + {(\tan \frac{α - β}{2})}^{2}}$
It’s obviously the too long to solve, so is there a shorter way to do this?

2alr8w 2022-01-25

Prove $\sin^{3} A - \cos^{3} A = (\sin^{2} A - \cos^{2} A) (1 - 2 \sin^{2} A \cos^{2} A)$

trefoniu1 2022-01-25

A trigonometry exercise $2 \sin \frac{π}{14} + 2 \sin \frac{5 π}{14} - 2 \sin \frac{3 π}{14} = 1$

Nylah Church 2022-01-24

If $\sin (x + 28^{\circ}) = \cos (3 x - 78^{\circ})$ , then find x

maliaseth0 2022-01-24

Solving $\sin (\arcsin (a - x) - c) = \sin$ for x $(\arcsin (b + x) + π - c) + d$

Swebaceacichegh 2022-01-24

I am practice about find the x as solution of
$\sin (\frac{2 x}{3} + 36^{\circ}) = \frac{12}{}$
Between $0^{\circ} and 360^{\circ}$ Here is my work so far,
$\sin (\frac{2 x}{3} + 36^{\circ}) = {\sin 30}^{\circ}$
I try using arcsin
$\frac{2 x}{3} + 36^{\circ} = 30^{\circ}$
$\frac{2 x}{3} = - 6^{\circ}$
$x = - 9^{\circ}$
I am stucked here.

Alex Cervantes 2022-01-24

Prove that $\cos e c^{2} A \cdot \sec^{2} A = \tan^{2} A + \cot^{2} A + 2$

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Trigonometry

Trigonometry Problems - Basics, Advanced & Examples