Solving Trigonometry Equations and Problems with Examples and Tips

If the solutions for $θ$ from the equation $\sin^{2} θ - 2 \sin θ + λ = 0$ lie in $\cup_{n \in z} (2 n π - \frac{π}{6}, (2 n + 1) π + \frac{π}{6})$ . Then find the possible set values of $λ$

Zugdichte2r 2022-01-26

Find all 4 complex roots to $3 z^{4} - z^{3} + 2 z^{2} - z + 3 = 0$ using a trig substitution.
A previous part of the question made us prove that $2 \cos n θ = z^{n} + z^{- n}$ and it wants us to prove that the polynomial can be expressed as $6 \cos 2 θ - 2 \cos θ + 2 = 0$ . I'm assuming that using $6 \cos 2 θ - 2 \cos θ + 2 = 0$ we will be able to find all the roots of the equation but I'm unable to see how to get the polynomial into the trig equation and then how to solve the trig equation.

Aubrey Hendricks 2022-01-26

Precalc Trig Identity, verify: $1 + \cos (x) + \cos (2 x) = \frac{12}{+} \frac{\sin (5 \frac{x}{2})}{2 \sin (\frac{x}{2})}$

Lizeth Jackson 2022-01-26

Evaluating $\sum_{n = 1}^{\infty} \frac{\sin (n x)}{n}$ without integrating $\sum_{n = 1}^{\infty} e^{n x}$
My solution proceeds by integrating
$\sum_{n = 1}^{\infty} e^{n x} = \frac{e^{i x}}{1 - e^{i x}}$

Kasey Haley 2022-01-26

Evaluate $lim_{x \to 0} \frac{1 - \cos x \cos 2 x \dots \cos n x}{x^{2}}$
It should be
$\frac{1}{12} n (n + 1) (2 n + 1)$
but I don't know how to prove that. I am also aware that $lim_{θ \to 0} \frac{1 - \cos θ}{θ^{2}} = \frac{1}{2}$ but I don't know how to use that.

Wendy Gutierrez 2022-01-26

calculating 2 constants in a function
$M = {f \in C [0, 2 π], \int_{0}^{2 π} f (x) \sin x, d x = π, \int_{0}^{2 π} f (x) \sin 2 x, d x = 2 π}$
$a, b \in R, g \in M, g (x) = a \sin x + b \sin 2 x, x \in [0, 2 π]$
Ive

Marissa English 2022-01-26

Showing that $f (4 x) + 4 f (2 x) = 8 \cos^{4} (x) - 3$ ,where $f (x) = \cos x$

Swebaceacichegh 2022-01-26

If $\tan (x_{1}) \dots \tan (x_{n}) = 1$ for acute $x_{i}$ , then does it follow that $\cos (x_{1}) + \dots + \cos (x_{n}) \leq n \frac{\sqrt{2}}{2}$ ?

Jett Castaneda 2022-01-26

Error evaluating $lim_{x \to 0} \frac{x - \tan x}{x^{3}}$
I solved it like this,
$lim_{x \to 0} (\frac{1}{x^{2}} - \frac{\tan x}{x^{3}}) = lim_{x \to 0} (\frac{1}{x^{2}} - \frac{\tan x}{x} \cdot \frac{1}{x^{2}})$
Now using the property
$lim_{x \to 0} \frac{\tan x}{x} = 1$
we have:
$lim_{x \to 0} (\frac{1}{x^{2}} - \frac{1}{x^{2}}) = 0$
Please explain my error! How can I avoid such errors?

Lizeth Jackson 2022-01-26

If $\cos θ = \frac{\cos α + \cos β}{1 + \cos α \cos β}$ , then prove that one value of $\tan (\frac{θ}{2})$ is $\tan (\frac{α}{2}) \tan (\frac{β}{2})$

Yahir Haas 2022-01-26

Evaluate $(\sum_{k = 1}^{7} \tan^{2} (\frac{k π}{16})) - \tan^{2} (\frac{4 π}{16})$

Emmy Combs 2022-01-26

Evaluate the sum of the infinite series $1 + \cos x + \cos^{2} x + \cos^{3} x \dots$ for $0 < x < π$
Therefore $\sum_{n = 0}^{\infty} \cos^{n} x$
which is just a geometric series with common ratio $\cos x$ and first term 1. So the sum of the series should be
$\sum_{n = 0}^{\infty} \cos^{n} x = \frac{1}{1 - \cos x}$
However, the answer to the question is ${\frac{12}{\csc}}^{2} (\frac{x}{2})$

siutulysr5 2022-01-26

What's the value of cos(x) that verifies the following equality
The equality is:
$2 \sin^{2} 2 x - 2 \sin x \sin 3 x = 4 \cos x + \cos 2 x$

Sereinserenormg 2022-01-26

Find the eccentricity of the conic given by:
$(x {\tan 10}^{\circ} + y {\tan 20}^{\circ} + {\tan 30}^{\circ}) (x {\tan 120}^{\circ} + y {\tan 220}^{\circ} + {\tan 320}^{\circ}) + 2018 = 0$

Jett Castaneda 2022-01-26

The absolute of the expression $\frac{1 + \sqrt{3} {\tan 1}^{\circ}}{\sqrt{3} - {\tan 1}^{\circ}} \cdot \frac{{\tan 89}^{\circ}}{{\cot 3}^{\circ}}$ is equal to...

sentidosin0q 2022-01-26

Given $\cos 3 θ = 4 {(\cos θ)}^{3} - 3 \cos θ$ , solve $4 x^{3} - 9 x - 1 = 0$ , correct to 3 decimal places.

Madilyn Fitzgerald 2022-01-26

Find: $\int_{0}^{\frac{π}{4}} x \ln (\sin x) d x$

Kristina Mcclain 2022-01-26

Why $\tan^{- 2} (x) = \frac{1}{\tan^{2} (x)}$ ?

Gerald Ritter 2022-01-26

If $x \in [- 1, 0)$ , then what is the value of $\cos^{- 1} (2 x^{2} - 1) - 2 \sin^{- 1} x$ ?

iocasq4 2022-01-26

If $\cos (a - b) = 3 \sin (a + b)$ then find $\frac{1}{1 - 3 \sin 2 a} + \frac{1}{1 - 3 \sin 2 b}$

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