Trigonometric Equations and Indentities: Real-World Examples

High School
Geometry
Trigonometry
Trigonometric equations and identities

Recent questions in Trigonometric equations and identities

Lucille Davidson 2022-01-16

Maximum and minimum of $f (x) = 2 \sin^{2} x + 2 \cos^{4} x$

Susan Nall 2022-01-16

$50 \cos^{2} x + 5 \cos x = 6 \sin^{2} x$ , find $\tan x$

hadejada7x 2022-01-16

$lim_{x \to 0} \frac{x (1 + a \cos x) - b \sin x}{x^{3}} = 1$ , how to find the constants a,b?

namenerk 2022-01-15

Sum of series of $\cot^{- 1} \frac{5}{\sqrt{3}} + \cot^{- 1} \frac{9}{\sqrt{3}} + \cot^{- 1} \frac{15}{\sqrt{3}} + \cot^{- 1} \frac{23}{\sqrt{3}} + \dots$

Carol Valentine 2022-01-15

Calculate the limit of $lim_{x \to 1} \frac{1 - x^{2}}{\sin π x}$ only with the fundamental limits

William Curry 2022-01-15

Solving the equation $\sin 5 x = 16 \sin^{5} x$

eiraszero11cu 2022-01-15

Limit $lim_{x \to 0} (\cot x - \frac{1}{\sin x})$

nemired9 2022-01-14

Proving that an $\int_{- \infty}^{\infty} \int_{k}^{k + 1} \sin (\exp (x)) d x d k$ integral converges

regatamin2 2022-01-03

Ive

Pamela Meyer 2022-01-03

How can I prove the identity
$\frac{1}{1 + \sin (x)} \equiv \frac{\sec^{2} (\frac{x}{2})}{{(\tan (\frac{x}{2}) + 1)}^{2}}$

compagnia04 2022-01-03

I have this identity:
$\cos^{2} 4 α = \cos^{2} 2 α - \sin 6 α \sin 2 α$
If I write this like as:
$(\cos 4 α + \cos 2 α) \cdot (\cos 4 α - \cos 2 α) = - \sin 6 α \sin 2 α$
I can use Werner and prostaferesis formulas and I find the identity (1).
But if we suppose of not to use these formulas I have done a try writing:
$\cos^{2} 4 α = \cos^{2} 2 α - \sin 6 α \sin 2 α$ (1)
${[\cos (2 (2 α))]}^{2} = {(2 \cos 2 α - 1)}^{2} - \sin 6 α \sin 2 α$ (2)
$\cos^{4} 2 α - 2 \sin^{2} 2 α \cos^{2} 2 α + \sin^{4} 2 α = {(2 \cos^{2} α - 1)}^{2} - \sin 6 α \sin 2 α$ (3)
Then I have abandoned because I should do long computations and I think that is not the right way.

aspifsGak5u 2022-01-03

$4 \sin (x) + 7 \cos (x) = 6$
where $0 \leq x \leq 360^{\circ}$
I put the equation into the form $a \sin (x) + b \cos (x) = R \sin (x + a)$ , but after determining that $R \cos (a) = 4, R \sin (a) = 7$ and $R \sin (x + a) = 6$ , I don't know how to proceed.

Paligutanhk 2022-01-03

How do you solve $5 \sin^{2} (x) + 8 \sin x \cos x - 3 = 0$ ?
I have tried using compound angle, sum to product, pythag identities but nothing seems to work. I tried turning it into $\sin 2 x$ but then I have $\sin^{2} x$ and $\sin 2 x$ together.

William Collins 2022-01-03

I tried it using the double angle identity
$2 \sin x \cos x$
The answer that I got is
$\frac{- \cos 2 x}{4} + c$
However Ive

killjoy1990xb9 2022-01-03

$\cos^{- 1} \frac{3}{\sqrt{10}} + \cos^{- 1} \frac{2}{\sqrt{5}} = ?$
Let $\cos^{- 1} \frac{3}{\sqrt{10}} = α, \cos^{- 1} \frac{2}{\sqrt{5}} = β$ then, $\cos α = \frac{3}{\sqrt{10}}, \cos β = \frac{2}{\sqrt{5}}$
Therefore
$\cos α = \frac{3 \cdot 2}{2 \sqrt{2} \sqrt{5}} = \frac{3}{2 \sqrt{2}} \cdot \cos β$

Russell Gillen 2022-01-03

Since $5 i = 5 e^{i \frac{π}{2}}$ And $2 + i = \sqrt{5} e^{i x_{1}}$ where $x_{1} = \arctan \frac{12}{}$ , we have $\frac{5 i}{2 + i} = \sqrt{5} [\cos (\frac{π}{2} - x_{1}) + i \sin (\frac{π}{2} - x_{1})]$ Now from here how do I continue?

Alan Smith 2022-01-03

How to prove $\frac{e^{j x} - j e^{- j x}}{j e^{j x} - e^{- j x}} = \frac{\tan x - 1}{\tan x + 1}$
where $j = \sqrt{- 1}$
Ive

Gregory Jones 2022-01-03

How can I prove that:
${(1 + \sin x + \cos x)}^{2} = 2 (1 + \cos x) (1 + \sin x)$
Ive

Carla Murphy 2022-01-02

The question is to prove:
$- \sqrt{a^{2} + b^{2} + 2 a b \cos (α - β)} < a \cos (α + θ) + b \cos (β + θ) < \sqrt{a^{2} + b^{2} + 2 a b \cos (α - β)}$
I tried opening the brackets but Im

Brock Brown 2022-01-02

How do I evaluate $lim_{z \to 1} (1 - z) \tan \frac{π z}{2}$
I tried using the identity, $\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x}$ to simplify this to:
$lim_{z \to 1} (1 - z) \frac{\sin π z}{1 + \cos π z}$

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Trigonometry

Demystify Trigonometric Equations and Identities with Step-by-Step Examples and Expert Guidance