Trigonometric Equations and Indentities: Real-World Examples

High School
Geometry
Trigonometry
Trigonometric equations and identities

Recent questions in Trigonometric equations and identities

Frankie Burnett 2022-11-02

Trigonometric equation with generalized solution
The equation
$\cot A - \tan A = 2$
Find the generalized form of A
I got the answer and the answer came
$\tan A = (2^{1 / 2}) - 1 and - 2^{1 / 2} - 1$
Now how to write the generalized form j here

caschaillo7 2022-10-31

How to solve the triangulation problem?
I have 3 sensors. I've built the following system of equations that match the data from sensors. I need to find
$x$ , $y$ , $R_{a}$ , $R_{b}$ , $R_{c}$ , $α$ , $β$ and $γ$ . Can you please help me, or at least hint how can I solve this system.
$\begin{aligned} A_{x} & = x + R_{a} \cos α \\ A_{y} & = y + R_{a} \sin α \\ B_{x} & = x + R_{b} \cos β \\ B_{y} & = y + R_{b} \sin β \\ C_{x} & = x + R_{c} \cos γ \\ C_{y} & = y + R_{c} \sin γ \\ R_{b} - R_{a} & = z_{1} \\ R_{c} - R_{a} & = z_{2} \end{aligned}$

snaketao0g 2022-10-31

Manipulating a trigonometric equation involving $\tan^{2} (3 θ)$
If $\tan^{2} 3 θ = 1$ , how do I manipulate the equation so I can make $\tan θ$ the subject? I forgot how to do these since it has been a long time. I tried searching before posting. My answer is $\frac{1}{\sqrt{3}}$ , not sure if it is correct though.

Marilyn Cameron 2022-10-31

Solving a trigonometric equation
Can someone help me to solve this problem?
Find all number pairs x,y that satisfy the equation:
$\tan^{4} (x) + \tan^{4} (y) + 2 \cot^{2} (x) \cot^{2} (y) = 3 + \sin^{2} (x + y)$

erwachsenc6 2022-10-30

Simplifying trigonometric equations
Here is the question: simplify the expression
$\frac{\sin (f + g) + \sin (f - g)}{\cos (f + g) + \cos (f - g)} .$
For this questions, are all of the addition and subtraction identities of sin and cos required? I am not certain how to approach this question. If someone could help it would be very appreciated!

Vincent Norman 2022-10-30

Finding the general solution for trigonometric equation
Solve for general solutions
$\tan (x / 3) = 1$
When I solve this equation my answer comes to be $x = 3 π / 4 \pm 2 n π, 15 π / 4 \pm 2 n π$ where $n$ is an integer
However when I graph the equation $y = \tan (x / 3) - 1$ values for $x$ such as $11 π / 4$ do not equal zero.
What would x be equal to then?

Yazmin Sims 2022-10-30

Complex Trigonometric Equation?
I am having difficulty solving
$\cos (z) = \sqrt{2}$
In class it was mentioned using the quadratic formula to solve for the inverse cos function but I am not sure how to do that.

Emilio Calhoun 2022-10-30

Help for this equation
Can anyone show me how to solve this trignometric equation :
$\sin (2 θ) - \cos (2 θ) = \frac{1}{2}$

Amina Richards 2022-10-30

Exponential trigonometrical equation
Find $x$ from $[- \frac{π}{2}, \frac{π}{2}]$ in:
$2^{\sin 3 x} - 8^{\sin x} = \sin^{3} x$
I know that $x = 0$ verifies the equation, but is it the only solution?

Brianna Schmidt 2022-10-29

Roots of trigonometric equation
In the following trigonometric equation
$1 + α^{2} \cos^{2} (n θ) = 0$
The complex solutions are
$\cos (n θ) = \pm i / α$
So I thought that the correspondant angles were
$n θ = \arccos (i / α) + 2 k π$
$n θ = 2 π - \arccos (i / α) + 2 k π$
and
$n θ = \arccos (- i / α) + 2 k π$
$n θ = 2 π - \arccos (- i / α) + 2 k π$
as usual in trigonometric equations containing cosine. But instead the solutions should be
$n θ = \arccos (i / α) + k π$
and
$n θ = \arccos (- i / α) + k π$
Why?

caschaillo7 2022-10-29

Uncertain how to solve this trigonometric equation
I am currently attempting to discover how to find the general solutions to
$\sqrt{3} \tan^{2} x = 2 \tan x + \sqrt{3}$
The given solutions are $x = \frac{π}{3} + π k$ , $\frac{5 π}{6} + π k$
To solve this equation I removed the square root from $\sqrt{3} \tan^{2} x$ leaving me with $\tan x$ . Then I subtracted $2 \tan x$ from $3 \tan x$ leaving me with $\tan x = \sqrt{3}$
Should I then not solve $\tan x = \sqrt{3}$ for the generalized solutions?
This would give $x = \frac{π}{3}$ and $x = \frac{2 π}{3}$ which would then be generalized to $\frac{π}{3} + π k$ and $\frac{2 π}{3} + π k$ , which doesn't agree with the given answers.

Jacoby Erickson 2022-10-28

Are these two equation equal? $f (x) = \sin (4 x) - \sin (3 x) \cos (x)$ and $g (x) = \sin (x) \cos (3 x)$

Mattie Monroe 2022-10-28

Finding the general solution of a trigonometric equation
Find the general solution of the equation
$2 \sin^{2} x + 5 \sin x \cos x - 8 \cos^{2} x = - 2$
My working:
$2 \sin^{2} x + 5 \sin x \cos x - 8 \cos^{2} x + 2 = 0$
$2 \sin^{2} x + 5 \sin x \cos x + 8 \sin^{2} x - 6 = 0$
$10 \sin^{2} x + 5 \sin x \cos x - 6 = 0$
$5 - 5 \cos 2 x + 5 \sin x \cos x - 6 = 0$
I am giving my working for the sake of it...I know I'm getting nowhere. I really need a hint as to how I can take this forward.

Kasey Reese 2022-10-28

Trigonometric equation with quadrangle
I need some help solving this. Any hints/ideas? We got a quadrangle, inscribed in a circle. The two opposite corners are $x$ and $y$ . (From this we know $x + y = 180^{\circ}$ )?
Find $\cos (x)$ if $\tan (y) = - 2$

independanteng 2022-10-28

How to solve this trigonometric equation?
I want to solve the equation
$(3 - \cos 4 x) (\sin x - \cos x) = 2.$
I solve by putting $t = \sin x - \cos x$ , but I can not find all solutions.

Diego Barr 2022-10-28

Finding cartesian equation from parametric trigonometric equations
I'm trying to find the cartesian equation of the curve which is defined parametrically by:
$x = 2 \sin θ, y = \cos^{2} θ$
Both approaches I take result in the same answer:
$y = 1 - \sin^{2} θ \sin θ = \sqrt{y - 1} x = 2 \sqrt{y - 1} x^{2} = 4 (y - 1) x^{2} + 4 = 4 y$
Method 2:
$\sin^{2} = y - 1 \sin θ = \frac{x}{2} \sin^{2} θ = \frac{x^{2}}{4} x^{2} + 4 = 4 y$
But the answer listed is $x^{2} + 4 y^{2} = 4$ . Are my calculations wrong?

Alessandra Cummings 2022-10-27

Establish the identity $\frac{\cos^{θ} - \sin^{2} θ}{1 - \cot^{2} θ} = - \sin^{2} θ$

miklintisyt 2022-10-27

Correct methodology for solving this trigonometric equation
I want the correct method for solving this trigonometric equation
$\sqrt{\cos (x) \sqrt{\cos (x) \sqrt{\cos (x) \dots}}} = 1.$
I have used my own technique:
$\begin{aligned} \sqrt{\cos (x) \sqrt{\cos (x) \sqrt{\cos (x) \dots}}} & = 1 \\ \cos^{\frac{1}{2}} (x) \cos^{\frac{1}{4}} (x) \cos^{\frac{1}{8}} (x) \dots & = 1 \\ \log (\cos^{\frac{1}{2}} (x) \cos^{\frac{1}{4}} (x) \cos^{\frac{1}{8}} (x) \dots) & = \log 1 \\ \log (\cos^{\frac{1}{2}} (x)) + \log (\cos^{\frac{1}{4}} (x)) + \log (\cos^{\frac{1}{8}} (x)) + \dots & = \log 1 \\ \frac{1}{2} \log (\cos (x)) + \frac{1}{4} \log (\cos (x)) + \frac{1}{8} \log (\cos (x)) + \dots & = \log 1 \\ \log (\cos (x)) (\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots) & = \log 1 \end{aligned}$
The series in the brackets is an infinite geometric series by summing it I get
$\begin{aligned} \cos (x) \frac{\frac{1}{2}}{1 - \frac{1}{2}} & = 1 \\ \cos (x) \frac{\frac{1}{2}}{\frac{1}{2}} & = 1 \\ \cos (x) & = 1 \\ x & = \arccos (1) \\ x & = 0 \end{aligned}$
So the solution set is ${2 n π}$ .
This is how I have solved the given problem but I want is there any other method for solving this equation.

Angel Kline 2022-10-27

Trigonometric equation, find $\sin θ$
Find $\sin θ$ if $a$ and $c$ are constants
$1 - {(c - a \tan θ)}^{2} = \frac{\sin^{2} θ \cos^{4} θ}{a^{2} - \cos^{4} θ}$

c0nman56 2022-10-26

Splitting a trigonometric equation into cases
I am not understanding this step in a solution I found. We are solving
$y = \frac{- 2 y}{1 - y^{2}}$
where $y = t a n (x)$
The solution says that we can now split this into two cases, case 1 where $y = 0$ and case 2 where $y \neq 0$ , I do not understand how we know that we can split up into two cases at this point, please advise. What's the motivation?
(As a side note, the cases are
case 1, y=0
$x = k π, k \in Z$
case 2, $y \neq 0$
$1 - y^{2} = - 2 ⟺ y^{2} = 3$
then
$x = \pm \frac{π}{3} + k π, k \in Z$ ).

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