Solving Trigonometry Equations and Problems with Examples and Tips

Solve the equation $\frac{\sqrt{3}}{2} \sin (x) - \cos x = \cos^{2} x$
My approach $\cos^{2} x = 1 - \sin^{2} x$
$\frac{\sqrt{3}}{2} \sin x - \cos^{2} x = \cos x$
$\frac{\sqrt{3}}{2} \sin x + \sin^{2} x - 1 = \cos x$
$\sqrt{3} \sin x + 2 \sin^{2} x - 2 = 2 \cos x$
Though the equation comes in form of $\sin x$ from here onward after squaring still not getting the answer.

Madilyn Fitzgerald 2022-01-30

How to simplify $(1 - \tan^{4} θ) \cos^{2} θ + \tan^{2} θ =$ ?
I have an answer, but Im

Rudy Koch 2022-01-30

Why $\sin (\frac{x}{y})$ is not equal to $\frac{\sin x}{\sin y}$ and why $\sin (x + y)$ is not equal to $\sin x + \sin y$
I get that this may be because they are functions, but what about the nature of trig functions causes the two to examples above to be not equal?
By the way, can you please keep the explanation very simple please? I am a high school student and may struggle to understand more complex explanations involving proof notation etc.

iocasq4 2022-01-30

How do I prove $x - \frac{x^{3}}{3} < \arctan x < x$ ?
for every x>0?
I tried taking the limit of $x - \frac{x^{3}}{3}$ and $\arctan (x)$ as x approaches 0, but I get 0 which makes sense since theyre

pozicijombx 2022-01-30

Laurent series of a fractional cosine and sine
i was wondering how i can write the laurent series of
$\cos (\frac{1}{z - 2})$
in z=2?
and if i want the laurent series of
$\sin (\frac{1}{z})$
(for Z=2 too , how it is because here 2 is not a singualrity and i dont

Gerald Ritter 2022-01-30

Evaluating $lim_{x \to 0} \frac{\cos x - \cos 3 x}{\sin 3 x^{2} - {\sin x}^{2}}$
Is there a simple way of finding the limit?
I know the long one: rewrite it as
$- lim_{x \to 0} \frac{\cos x - \cos (3 x)}{\sin (3 x^{2})} \cdot \frac{1}{1 - \frac{\sin (3 x^{2})}{\sin (x^{2})}}$
and then find both limits in separately applying LHospitals rule several times. The answer is 2

Marissa English 2022-01-30

Consider the series
$\sum_{n = 1}^{\infty} \log (1 + \frac{1}{| \sin (n) |})$
Determine whether it converges absolutely or conditionally.
I am trying to apply Cauchy condensation test, but I am not sure whether the given series is non-increasing or not.

derlingasmh 2022-01-30

Proving identity $\cos^{2} a = {sinh}^{2} b$ if $\sin (a + i b) = \cos b + i \sin b$

mairie0708zl 2022-01-30

How to solve $\cos (θ + ∠)$ equation?
I cant

Haialarmz6 2022-01-30

Solve $8 \sin x = \frac{\sqrt{3}}{\cos x} + \frac{1}{\sin x}$
My approach is as follow
$8 \sin x - \frac{1}{\sin x} = \frac{\sqrt{3}}{\cos x}$
On squaring we get
$64 \sin^{2} x + \frac{1}{\sin^{2} x} - 16 = \frac{3}{\cos^{2} x}$
$(64 \sin^{4} x - 16 \sin^{2} x + 1) (1 - \sin^{2} x) = 3 \sin^{2} x$
Solving and re-arranging we get $- 64 \sin^{6} x + 80 \sin^{4} x - 20 \sin^{2} x + 1 = 0$
Using the substitution $\sin^{2} x = t$
$- 64 t^{3} + 80 t^{2} - 20 t + 1 = 0$
I am not able to solve it from hence forth

bistandsq 2022-01-30

Find the general solution (i) $\cot θ = - d \frac{1}{\sqrt{3}}$
(ii) $4 \cos^{2} θ = 1$

Willow Carrillo 2022-01-30

Find $λ$ such that f(x) has no point of local maxima or minima in $| x | < \frac{π}{2}$
Let
$f (x) = {(\sin (x))}^{3} + λ {(\sin (x))}^{2}$
for every x in domain of $(\frac{- π}{2}, \frac{π}{2})$ has no point of local maxima or minima, then find the value of $λ$ for the given condition to follow.
1): found f′(x) and equated with zero to find that it only matters by $(3 \sin x + 2 λ)$ and sinx to have change in the sign of derivative around zero.So the $λ$ becomes 0 and 0 becomes the point of inflection.

treetopssan 2022-01-30

Let $z = 2 e^{10 π \frac{i}{21}} and w = e^{π \frac{i}{7}}$ . Then if $[{(z - w)}^{6} = r e^{i θ}]$ where $r \geq 0$ and $0 \leq θ \leq 2 π$ what is the ordered pair $(r, θ)$ ?

plorinosakl 2022-01-30

Is $\sin 2 x \geq 0$ same as $\sin x \cos x \geq 0$ ?

Madilyn Fitzgerald 2022-01-30

Given $\sin x = \frac{3}{4}$ and $\frac{π}{2} < x < π$ , find $\sin (x + \frac{π}{6})$
I have been trying to calculate this question
$\sin θ = \frac{3}{4}$
$\frac{π}{2} < θ < π$
$\sin (θ + \frac{π}{6}) = \sin θ . \frac{\cos π}{6} + \cos θ . \frac{\sin π}{6}$
$= \frac{3}{4} . \frac{\sqrt{3}}{2} + \cos θ \frac{.1}{2}$
$= \frac{3 . \sqrt{3} + 4 \cos θ}{8}$
The answer is: $\frac{3 . \sqrt{3} - \sqrt{7}}{8}$
However I do not know how to get to it.

Clare Baldwin 2022-01-30

Equation of line passing through (1,2) and perpendicular to the line 2x+3y=1.
i have attempted so far
to find a slope of line
2x+3y=1
Slope m of a line of the form Ax+By=C equals $- \frac{A}{B}, A = 2, B = 3$
$m = - \frac{23}{}$
the point slope formula
$y - y_{1} = m (x - x_{1})$
the point given (1,2)
$y - 2 = - \frac{23}{x - 1}$
$y = - \frac{23}{x} + \frac{23}{+} \frac{21}{\times} \frac{33}{}$
$y = - \frac{23}{x} + \frac{23}{+} \frac{63}{}$
$y = \frac{23}{x} + \frac{83}{}$
is this right answer ?

Connor Randall 2022-01-29

How to prove the following equation:
$\tan^{2} (\frac{π}{16}) + \tan^{2} (\frac{3 π}{16}) + \tan^{2} (\frac{5 π}{16}) + \tan^{2} (\frac{7 π}{16}) = 28$

Jayleen Sanders 2022-01-29

Atypical way to find angle between unit vectors: $θ = 2 \sin^{- 1} (\frac{1}{2} | \hat{A} - \hat{B} |)$
At my work, I have come across code with the following way of calculating the angle between two vectors.
$θ = 2 \sin^{- 1} (\frac{1}{2} | \hat{A} - \hat{B} |)$
(Note the physics convention: $\hat{v}$ indicates the normalization of v; ie, $\hat{v} = \frac{v}{| v |}$ ).

Ethen Wong 2022-01-29

How do you integrate $\int \frac{\cos (4 x)}{\cos (x)} d x$
I tried using trigonometric formulas for turning it into $2 \int \frac{\cos^{2} (2 x)}{\cos (x)} d x - \int \frac{1}{\cos (x)} d x$ and can solve the second one, but still no idea of how to proceed with $\int \frac{\cos^{2} (2 x)}{\cos (x)} d x$ .

ardeinduie 2022-01-29

If $\cos^{6} (x) + \sin^{4} (x) = 1$ find x if $x \in [0, \frac{π}{2}]$
My attempt:
$\cos^{6} (x) + \sin^{4} (x) = 1$
$\cos^{6} (x) + {(1 - \cos^{2} (x))}^{2} = 1$
$\cos^{6} (x) + 1 - 2 \cos^{2} (x) + \cos^{4} (x) = 1$
$\cos^{6} (x) + \cos^{4} (x) - 2 \cos^{2} (x) = 0$

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