Solving Trigonometry Equations and Problems with Examples and Tips

Is a function periodic $f (x) = \cos (x) + \cos (x^{2})$
This one I gave my students today, nobody solve it.
Is a following function periodic $f : R \to R$
$f (x) = \cos (x) + \cos (x^{2})$
If someone is interested I can show a solution later.

Halle Marsh 2022-04-24

Inverse Trigonometry System of Equations
$2 \tan^{- 1} (\sqrt{x - x^{2}}) = \tan^{- 1} (x) + \tan^{- 1} (1 - x)$
I have a feeling solution includes drawing triangles but cannot make the leap to get the solution

topoit8d 2022-04-24

Integrate the function
$\frac{2 x^{3} + 5 x^{2} + 8 x + 4}{{(x^{2} + 2 x + 2)}^{2}}$

dooporpplauttssg 2022-04-24

Integrate $\tan^{2} (\frac{π}{12} \cdot y)$

luminarc24lry 2022-04-24

In any triangle, if $\frac{\cos A + 2 \cos C}{\cos A + 2 \cos B} = \frac{\sin B}{\sin C}$ , prove that the triangle is either isosceles or right angled.
My Attempt:
Given:
$\frac{\cos A + 2 \cos C}{\cos A + 2 \cos B} = \frac{\sin B}{\sin C}$
$\frac{\frac{b^{2} + c^{2} - a^{2}}{2 b c} + 2 \frac{a^{2} + b^{2} - c^{2}}{2 a b}}{\frac{b^{2} + c^{2} - a^{2}}{2 b c} + 2 \frac{a^{2} + c^{2} - b^{2}}{2 a c}} = \frac{b}{c}$
On simplification,
$\frac{a b^{2} + a c^{2} - a^{3} + 2 a^{2} c + 2 b^{2} c - 2 c^{3}}{a b^{2} + a c^{2} - a^{3} + 2 a^{2} b + 2 b c^{2} - 2 b^{3}} = \frac{b}{c}$

Esther Hoffman 2022-04-24

In a $△ A B C, \frac{a}{b} = 2 + \sqrt{3}$ and $∠ C = 60^{\circ}$ Then find the ordered pairs $(∠ A, ∠ B) =$

gaitaprepeted05u 2022-04-24

Im trying to show that
$z = \tan [\frac{1}{i} \log (\sqrt{\frac{1 + i z}{1 - i z}})]$

predorsomwy 2022-04-24

If $y = - e^{x} \cos 2 x$ , show that $\frac{d^{2} y}{d x^{2}} = 5 e^{x} \sin (2 x + α)$ where $α = \tan^{- 1} (\frac{3}{4})$

Kaylee Torres 2022-04-24

If $2 \tan A = 3 \tan B$ then prove that
$\tan (A - B) = \frac{\sin 2 B}{5 - \cos 2 B}$

Naomi Hopkins 2022-04-24

If $11 γ = π$ then prove that $\sec (γ) \sec (2 γ) \sec (3 γ) \sec (4 γ) \sec (5 γ) = 32$

Esther Hoffman 2022-04-24

How many solutions to $z^{6!} - z^{5!} \in R and | z | = 1$
The following is my interpretation of the first solution.
The next step is to see how many solutions to $\sin (720 θ) - \sin (120 θ) = 0$ . (1)
One way to do this is consider $\sin (6 ω) - \sin (ω)$ , which has 12 solutions. (2)
For each solution to (2), corresponds 120 solutions to (1). I understand that.
But how do I know that $\sin (6 ω) = \sin (ω)$ has 12 solutions?

komanijuuxw 2022-04-23

Please prove $\frac{1 + \sin θ - \cos θ}{1 + \sin θ + \cos θ} = \tan (\frac{θ}{2})$

Lymnmeatlypamgfm 2022-04-23

In which point the tangent of $y = \sin x - \cos x$ is parallel with y=x?

Angeline Hayden 2022-04-23

If $z = \tan (\frac{x}{2})$ , show that $\sin (x) = \frac{2 z}{1 + z^{2}}$

broggesy9 2022-04-23

If $A + B + C = 180^{\circ}, \cot A = 2$ and $\cot B = \frac{3}{4}$ , what is $\sin C$ ?

Elise Winters 2022-04-23

if $a \cos θ - b \sin θ = c$
Prove that $a \cos θ - b \sin θ = \pm \sqrt[()]{a^{2} + b^{2} - c^{2}}$

Jakayla Benton 2022-04-23

If $2 \tan^{2} x - 5 \sec x = 1$ has exactly 7 distinct solutions for $x \in [0, \frac{n π}{2}], n \in N$ , then the greatest value of n is?
My attempt:
Solving the above quadratic equation, we get $\cos x = \frac{1}{3}$
The general solution of the equation is given by $\cos x = 2 n π \pm \cos^{- 1} \frac{1}{3}$
For having 7 distinct solutions, n can have value = 0,1,2,3
So, from here we can conclude that n is anything but greater than 6. So, according to the options given in the questions, the greatest value of n should be 13. But the answer given is 14. Can anyone justify?

airesex2 2022-04-23

Using Leibniz on $\sum_{n = 1}^{\infty} \sin (π \sqrt{n^{2} + 1})$

Aaron Coleman 2022-04-23

If $x_{n} = \cos {\sqrt{n + 1}} - \cos {\sqrt{n}}$ , what is $lim_{x \to \infty} x_{n}$ ?
I think it may be 0, because $\sqrt{n + 1}$ and $\sqrt{n}$ are two very close angles. But I don't know how to prove it. If I use the formula for $\cos - \cos$ , I get $- 0 \cdot \infty$

wuntsongo0cy 2022-04-23

Is it possible to solve $\cos (x) + 2 e^{x} = 0$ analytically?

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