Solving Trigonometry Equations and Problems with Examples and Tips

Use $e^{i a} + e^{i b}$ to show that $y (t) = 2 A \cos (\frac{δ}{2} t - \frac{ϕ_{1} - ϕ_{2}}{2}) \sin ((ω + \frac{δ}{2}) t + \frac{ϕ_{1} + ϕ_{2}}{2})$
Given formula (1)
$e^{i a} + e^{i b} = 2 e^{i \frac{(a + b)}{2}} \cos (\frac{a - b}{2})$
Formula (1) should be used to show
$y (t) = 2 A \cos (\frac{δ}{2} t - \frac{ϕ_{1} - ϕ_{2}}{2}) \sin ((ω + \frac{δ}{2}) t + \frac{ϕ_{1} + ϕ_{2}}{2})$

Malik Dean 2022-04-08

Use condition $\sin θ - \cos θ = \sqrt{2}$
Please find the value of $\frac{1}{\sin^{10} θ} + \frac{1}{\cos^{10} θ}$

Ashleigh Mitchell 2022-04-08

Trigonometry problem $\frac{{\sin 100}^{\circ} + {\cos 70}^{\circ}}{{\cos 80}^{\circ} - {\cos 20}^{\circ}}$
I've done trigonometry in my earlier years of high school but I forgot a lot of rules. This is where I'm stuck on this problem:
$\frac{{\sin 100}^{\circ} + {\cos 70}^{\circ}}{{\cos 80}^{\circ} - {\cos 20}^{\circ}} =$
$\frac{\sin (90^{\circ} + 10^{\circ}) + \cos (60^{\circ} + 10^{\circ})}{\cos (90^{\circ} - 10^{\circ}) - \cos (30^{\circ} + 10^{\circ})} =$
$\frac{{\sin 90}^{\circ} {\cos 10}^{\circ} + {\cos 90}^{\circ} {\sin 10}^{\circ} + {\cos 60}^{\circ} {\cos 10}^{\circ} - {\sin 60}^{\circ} {\sin 10}^{\circ}}{{\cos 90}^{\circ} {\cos 10}^{\circ} + {\sin 90}^{\circ} {\sin 10}^{\circ} - {\cos 30}^{\circ} {\cos 10}^{\circ} + {\sin 30}^{\circ} {\sin 10}^{\circ}} =$
$\frac{{\cos 10}^{\circ} + \frac{1}{2} {\cos 10}^{\circ} - \frac{\sqrt{3}}{2} {\sin 10}^{\circ}}{{\sin 10}^{\circ} - \frac{\sqrt{3}}{2} {\cos 10}^{\circ} + \frac{1}{2} {\sin 10}^{\circ}} =$
$\frac{\frac{3}{2} {\cos 10}^{\circ} - \frac{\sqrt{3}}{2} {\sin 10}^{\circ}}{\frac{3}{2} {\sin 10}^{\circ} - \frac{\sqrt{3}}{2} {\cos 10}^{\circ}}$

zdebe5l8 2022-04-08

Trigonometry Identity:
$3 (\cot^{2} θ + 1) - \csc^{2} θ - 1 = \cot^{2} θ + \csc^{2} θ$

painter555ui8n 2022-04-08

Trigonometric Property when a+b+c=abc
then $\exists α, β, γ$ $a = \tan α; b = \tan β; c = \tan γ$ where $α + β + γ = π$
I don't know how to prove it

Malachi Mullins 2022-04-08

Trigonometric equations with 2 functions
Given the equation: $\sin^{2} x + \cos x = 0$

Briley Cabrera 2022-04-08

Tips on evaluating $lim_{x \to 0} \frac{\sqrt{1 - \cos (x^{2})}}{1 - \cos x}$ ?

Magdalena Warren 2022-04-08

The value of the expression $x^{4} - 8 x^{3} + 18 x^{2} - 8 x + 2$ when $x = \cot \frac{π}{12}$

Efrain Watkins 2022-04-07

For how many values of x in $0 \leq x < 2 π$ does $\frac{\sin (5 x)}{\sin (x)} - \frac{\cos (5 x)}{\cos (x)} = 2$
Using the sine and cosine addition rules, I tried to express $\sin (5 x)$ and $\cos (5 x)$ in terms of $\sin (x)$ and $\cos (x)$ . However, it took me quite a while, and I was wondering if there was a quicker way

Aedan Berry 2022-04-07

Fixing $γ, γ^{'}$ two reals numbers such that
$| γ^{'} | < \frac{1}{2} and 0 < γ < 1$ ,
and $α, β \in (0, π)$ , define
$x_{n} = \sin (2 π n γ + α + β) + \sin (2 π n γ^{'} + α - β)$
Could someone give me a simple statement equivalent to $x_{n} \geq 0$

Camila Glenn 2022-04-07

Finding $lim_{x \to 0} \frac{\sin (π \cos^{2} (\frac{x}{2}))}{\sin (\sin (x))}$
I have the following solution, only the first equality of which bothers me:
$lim_{x \to 0} \frac{\sin (π \cos^{2} (\frac{x}{2}))}{\sin (\sin (x))} = lim_{x \to 0} \frac{\sin (π \sin^{2} (\frac{x}{2}))}{\sin (\sin (x))} = lim_{x \to 0} π \frac{\sin (\frac{x}{2})}{2 \cos (\frac{x}{2})} . \frac{2 \sin (\frac{x}{2}) \cos (\frac{x}{2})}{\sin (2 \sin (\frac{x}{2}) \cos (\frac{x}{2}))} . \frac{\sin (π \sin^{2} (\frac{x}{2}))}{π \sin^{2} (\frac{x}{2})} = 0$
What is the justification for replacing $\cos^{2}$ with $\sin^{2}$ ?

Quinn Dean 2022-04-07

Finding $\frac{1}{1 + {\tan 70}^{\circ}} + \frac{1}{1 + {\tan 20}^{\circ}}$

George Michael 2022-04-07

Find:
$lim_{x \to 0} \frac{1 - \cos 4 x}{x \tan 6 x}$

Rosa Townsend 2022-04-07

Find value of
$\prod_{r = 1} \sin (\frac{(2 r - 1) π}{2 n})$
Where $n \in N$ and $n > 1$

Aliyah Mendez 2022-04-07

find the x: $(\cos 3 x + \cos 4 x) (\cos 3 x + \cos x) = \frac{1}{4}$

Colt Rhodes 2022-04-07

Find the values of ${\sin 18}^{\circ}$

delitzo1d4 2022-04-07

Find the value of the following series.
$\tan θ + 2 \tan (2 θ) + 2^{2} \tan (2^{2} θ) + \dots + 2^{14} \tan (2^{14} θ) + 2^{15} \cot (2^{15} θ)$

Aileen Rogers 2022-04-07

Find the value of $\cos (θ)$ given $\cot (θ)$
$23 \frac{\sqrt{530}}{530}$ but I keep getting $\cos (θ) = \frac{1}{\sqrt{530}}$ . Can someone please tell me what I’m doing wrong?
$1 + \cot^{2} (θ) = \frac{1}{\cos^{2} (θ)}$ // Pythagorean identity
$1 + 529 = \frac{1}{\cos^{2} (θ)}$ // Substitute $\cot^{2} (θ)$
$530 = \frac{1}{\cos^{2} (θ)}$
$530 \cos^{2} (θ) = 1$ // Divide both sides by $\cos^{2} (θ)$
$\cos^{2} (θ) = \frac{1}{530}$ // Divide both sides by 530
$\cos (θ) = \frac{1}{\sqrt{530}}$ // Square root both sides

Miguel Hanson 2022-04-07

Find the value of $10 \sin^{4} θ + 15 \cos^{4} θ = 6$

Boston Fitzgerald 2022-04-07

Find the value of $\cot (16) \cot (44) + \cot (44) \cot (76) - \cot (76) \cot (16)$

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