Solving Trigonometry Equations and Problems with Examples and Tips

Show that $5 \cos^{2} x - 2 \sqrt{3} \sin x \cos x + 3 \sin^{2} x = \cos 2 x - \sqrt{3} \sin 2 x + 4$

Eileen Garza 2022-05-03

Show that
$\frac{\sin (3 x)}{\sin x} + \frac{\cos (3 x)}{\cos x} = 4 \cos (2 x)$

Dereon Guzman 2022-05-03

Show that $\forall x \in R$ $\cos (\cos x) > \sin (\sin x)$
I've tried the obvious thing to do:
$φ : x \mapsto \cos (\cos x) - \sin (\sin x)$
$φ^{'} (x) = \sin (\cos x) \sin x - \cos (\sin x) \cos x$
I'd like to show now that $φ$ is a positive function for all $x \in R$ , but the derivative does not look friendly so either there's something with $φ^{'}$ . I'm not seeing or it's not the right way. Any hint or idea?

ulcerat9jr 2022-05-03

Show that ${\sin 10}^{\circ}$ is irrational.

Treboldiqtw 2022-05-03

Show that
$- 2 \leq \cos θ (\sin θ + \sqrt{\sin^{2} θ + 3}) \leq 2$
for all value of $θ$ .

luminarc24lry 2022-05-02

Simple trigonometry
$\frac{{\sin 100}^{\circ}}{{\sin 60}^{\circ}} + \frac{1}{1 + 2 {\sin 50}^{\circ}} = 2 {\sin 50}^{\circ}$

Maximillian Patterson 2022-05-02

Showing that if $x = \sec θ + \tan θ$ ,then $x + \frac{1}{x} = 2 \sec θ$
I used a few simple trig identities but get nowhere. I am confused about this question a lot. I do not see where the theta and x can be in one.
None are squared so I cannot used the identities I would think. The only thing I can think to do is $x = \frac{1}{\sin θ} + \frac{\cos θ}{\sin θ}$

Deven Livingston 2022-05-02

Showing surjectivity of $f (x) = \frac{x - \sin x}{1 - \cos x}$
is strictly monotonically increasing on $x \in (0, 2 π)$ , I'd like to show the surjectivity of this function [to $(0, \infty)$ ]. However, I fail solving the equation
$y = \frac{x - \sin (x)}{1 - \cos (x)}$
with respect to x. Can I somehow use/apply the fact that $f (x) = \frac{u (x)}{u^{'} (x)}$ ?

Slade Higgins 2022-05-01

Show that ${| \frac{(1 - α) (1 + {(e^{j w})}^{- 1})}{2 (1 - α {(e^{j w})}^{- 1})} |}^{2} = \frac{{(1 - α)}^{2} (1 + \cos (w))}{2 (1 + α^{2} - 2 α \cos (w)}}$
So far, I have taken $| H_{L P} (e^{j ω}) | = \frac{1 - α}{2} \frac{1 + \cos (ω)}{1 - α \cos (ω)}$
Squaring it subsequently gives $\frac{{(1 - α)}^{2}}{4} \frac{{(1 + \cos (ω))}^{2}}{1 - 2 α \cos (ω) + α^{2} \cos^{2} (ω)}$

Wertgenib6 2022-05-01

Show that:
$2^{\sin x} + 2^{\cos x} \geq 2^{(1 - \frac{1}{\sqrt{2}})}$

tefoldaavv 2022-05-01

Show that the equation below is only possible when $x = y$
$\sec^{2} θ = \frac{4 x y}{{(x + y)}^{2}}$

ahaar412rs7 2022-05-01

Show that $\sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos 8 θ}}} = 2 \cos θ$

etudiante9c2 2022-05-01

Show that $\sin (2 x) + \sin (4 x) + . . . + \sin (2 n x) = \frac{\sin (n x) \sin ((n + 1) x}{\sin (x)}$

Taliyah Spencer 2022-05-01

Show that ${(1 - \cos θ - \sin θ)}^{2} - 2 (1 - \sin θ) (1 - \cos θ) = 0$

ulcerat9jr 2022-05-01

Show that
$\sin (x) = x \prod_{n = 1}^{\infty} (1 - \frac{4}{3} \sin^{2} (\frac{x}{3^{n}}))$
I know $1 - \sin^{2} x = (1 - \sin x) (1 + \sin x)$ but I'm not sure how to apply it.
Or maybe we need $2 \sin^{2} (x) = 1 - \cos (2 x)$ ?

luminarc24lry 2022-04-30

Showing that $\int_{0}^{\frac{π}{2}} \cos^{2 n + 1} (x) d x = \frac{4^{n} {(n!)}^{2}}{(2 n + 1)!}$ ?

Gage Potter 2022-04-30

Simplify
$7 \arctan^{2} ϕ + 2 \arctan^{2} ϕ^{3} - \arctan^{2} ϕ^{5}$

Kailey Perez 2022-04-30

Show: $\frac{\cos 3 x + i \sin 3 x}{\cos x + i \sin x} = \cos 2 x + i \sin 2 x$

Elise Winters 2022-04-30

Show that $\sin (\frac{π}{3} (x - 2))$ is equal to $\cos (\frac{π}{3} (x - \frac{7}{2}))$

Maximillian Patterson 2022-04-30

Show that $\sin (\frac{x}{2}) + 2 \sin (\frac{x}{4}) \leq 3 \frac{\sqrt{3}}{2}$ for all $x \in [0, 2 π]$
My try
$f (x) = \sin (\frac{x}{2}) + 2 \sin (\frac{x}{4})$
$f^{'} (x) = \frac{1}{2} \cos (\frac{x}{2}) + \frac{1}{2} \cos (\frac{x}{4})$
$f^{'} (x) = \frac{1}{2} \cos (\frac{x}{2}) + \frac{1}{2} \cos (\frac{x}{4})$
$f^{'} (x) = \frac{1}{2} (\cos (\frac{x}{2}) + \cos (\frac{x}{4}))$
$f^{'} (x) = \cos (3 \frac{x}{4}) \cos (\frac{x}{4}))$
$f^{'} (x) = 0 \Rightarrow \cos (3 \frac{x}{4}) \cos (\frac{x}{4})) = 0$
$\cos (3 \frac{x}{4}) = 0 or \cos (\frac{x}{4}) = 0$
$3 \frac{x}{4} = \frac{π}{2} and so x = 2 \frac{π}{3} or x = 2 π$
$\frac{x}{4} = \frac{π}{2} and so x = 2 π$
Is my work ok ? What should I do now?

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