Trigonometric Equations and Indentities: Real-World Examples

High School
Geometry
Trigonometry
Trigonometric equations and identities

Recent questions in Trigonometric equations and identities

rzfansubs87 2022-07-07

Trigonometry equation $\tan 2 x + 3 \sec x + 3 = 0$ for $\tan 2 x + 3 \sec x + 3 = 0$

invioor 2022-07-06

Trig Identities Question: $\frac{\sin (4 t) (1 - \cos (2 t))}{\cos (2 t) (1 - \cos (4 t))} = \tan t, 0 < t < \frac{π}{2}, t \neq \frac{π}{4} .$

Montenovofe 2022-07-06

I have this equation:
$x^{2} + (x - 7)^{2} = 13^{2}$
Why can't squaring the whole equation be used to solve it?
$\sqrt{x^{2} + (x - 7)^{2} = 13^{2}} \equiv x + x - 7 = 13$

rzfansubs87 2022-07-05

Is there an inequality for $\sinh (x)$ which is similar to this inequality $\cosh (x) \leq e^{x^{2} / 2}$

Raul Walker 2022-07-04

Trigonometric Identify: $\frac{1 + \cos x}{\sin x} = \cot \frac{x}{2}$

kramberol 2022-07-04

How to Solve for x in the equation $\sin (2 x) = - \frac{1}{2}$ within the interval of $[0, 2 π]$ ?

veneciasp 2022-07-04

Solve the equation $\frac{1}{\sin^{2 k} (x)} + \frac{1}{\cos^{2 k} (x)} = 8$ where k is an integer and x is a real number.

mistergoneo7 2022-07-03

Find the solution set of the equation $5. (\frac{1}{25})^{\sin^{2} x} + {4.5}^{\cos 2 x} = 25^{\frac{\sin 2 x}{2}}$ where $x \in [0, 2 π]$
My approach :
$5. (\frac{1}{25})^{\sin^{2} x} + {4.5}^{1 - 2 \sin^{2} x} = 25^{\frac{\sin 2 x}{2}}$
Unable to understand how to use $\sin 2 x$ in R.H.S. to solve further , can you please guide further thanks

rzfansubs87 2022-07-02

trigonometry equation, minus problem, $\sin x + \sin 5 x = 0$
in the book , there are 2 solutions: x=60k , x=45+90k. in my solution, I got the first solution right, but the second solution with a minus. I don't understand what I did wrong.
$\sin (x) + \sin (5 x) = 0 \sin (x) = - \sin (5 x) \sin (x) = \sin (- 5 x)$
$x = - 5 x + 360 k 6 x = 360 k x = 60 k$
$x = 180 - (- 5 x) + 360 k x = 180 + 5 x + 360 k - 4 x = 180 + 360 k x = - 45 - 90 k$

hryggcx 2022-07-02

Any ideas on how to solve this trig equation: $\tan (π / 4 + x) = 3 \tan (π / 4 - x)$ ?

Sonia Ayers 2022-07-01

If $x = r \cos (θ)$ then in $? = r \cos (θ + a)$ what is ? equal to?

vasorasy8 2022-07-01

Proving this identity: $\frac{\sin^{2} θ}{\cos^{2} θ} + \frac{\cos^{2} θ}{\sin^{2} θ} = \sec^{2} θ \csc^{2} θ - 2$

enrotlavaec 2022-06-27

Establishing the identity $\frac{\sin (α + β)}{\cos α \cos β} = \tan α + \tan β$

Dale Tate 2022-06-26

trying to solve $\sqrt{\cos (x) - 2 \cos (2 x)} + \sqrt{2} \cos (2 x) = 0$
The system is
${\begin{cases} \cos (x) - 2 \cos (2 x) = 2 \cos^{2} (2 x) \\ - \sqrt{2} \cos (2 x) \geq 0 ⟺ \cos (2 x) \leq 0 \end{cases}$
The equation:
$\cos (x) - 2 (2 \cos^{2} (x) - 1) = 2 (2 \cos^{2} (x) - 1)^{2}$
$\cos (x) - 4 \cos^{2} (x) + 2 = 2 (4 \cos^{4} (x) - 4 \cos^{2} (x) + 1)$
$\cos (x) - 4 \cos^{2} (x) + 2 = 8 \cos^{4} (x) - 8 \cos^{2} (x) + 2$
$8 \cos^{4} (x) - 4 \cos^{2} (x) - \cos (x) = 0$
But what to do next?
If I factor out $\cos (x)$ , I get a non-factorizable third-degree polynomial in parentheses
$\cos (x) (8 \cos^{3} (x) - 4 \cos (x) - 1) = 0$

dourtuntellorvl 2022-06-25

Prove the following:
$(\binom{n}{0}) - (\binom{n}{2}) + (\binom{n}{4}) - (\binom{n}{6}) + . . . = 2^{\frac{n}{2}} \cos \frac{n π}{4}$

pachaquis3s 2022-06-25

I needed to solve the following equation:
$\tan θ + \tan 2 θ + \tan 3 θ = \tan θ \tan 2 θ \tan 3 θ$
Transform the LHS first:
$\begin{aligned} \tan θ + \tan 2 θ + \tan 3 θ & = (\tan θ + \tan 2 θ) + \frac{\tan θ + \tan 2 θ}{1 - \tan θ \tan 2 θ} \\ = \frac{(\tan θ + \tan 2 θ) (2 - \tan θ \tan 2 θ)}{1 - \tan θ \tan 2 θ} \end{aligned}$
And, RHS yields
$\begin{aligned} \tan θ \tan 2 θ \tan 3 θ & = (\tan θ \tan 2 θ) \frac{\tan θ + \tan 2 θ}{1 - \tan θ \tan 2 θ} \end{aligned}$
Now, two terms can be cancelled out from LHS and RHS, yielding the equation:
$\begin{aligned} 2 - \tan θ \tan 2 θ & = \tan θ \tan 2 θ \\ \tan θ \tan 2 θ & = 1, \end{aligned}$
which can be further reduced as:
$\tan^{2} θ = \frac{1}{3} ⟹ \tan θ = \pm \frac{1}{\sqrt{3}}$
Now, we can yield the general solution of this equation:
$θ = n π \pm \frac{π}{6}, n \in Z$ . But, setting $θ = \frac{π}{6}$ in the original equation is giving one term $\tan \frac{π}{2}$ , which is not defined.

polivijuye 2022-06-25

Is there a way to prove this relation?:
$\sum_{n = 0}^{\infty} \frac{1}{s^{2} + {(1 + 2 n)}^{2} ω^{2}} = \frac{π \tanh (\frac{π s}{2 ω})}{4 s ω}$
And find the conditions for which this equality hold?

hawatajwizp 2022-06-24

I am trying to show that for every $λ \in (- π, π) ∖ {0}$ and every $n \in Z_{> 0}$
$\frac{1}{n} \sqrt{\frac{1 - \cos (n - 1) λ}{1 - \cos λ}} \leq 1$

Adriana Ayers 2022-06-24

How would I solve the following trigonometric equation?
$2 \cot 2 x \cos 2 x = 1 - \sin 2 x$
I got to this stage: $2 \cos^{2} 2 x = \sin 2 x - \sin^{2} 2 x$
How do I continue?

Emmy Dillon 2022-06-24

Prove that
$3 x - x^{3} < \frac{2}{\sin 2 x}, \forall x \in (0, \frac{π}{2})$
I have tried by proving that
$3 x - x^{3} < \frac{9}{5 π} x + \frac{3}{2} < \frac{2}{\sin 2 x}, \forall x \in (0, \frac{π}{2})$
with Jensen's inequality, but I hoped this problem would have a simpler solution.

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Trigonometry

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