# Trigonometry reflection problem

• Lindsayyyy
In summary, the conversation revolves around solving a trigonometric problem involving finding the value of alpha. The original poster is stuck at one step and is seeking help from others. One person suggests using an approximation method while the other provides a detailed approach to solving the problem. They also mention an extra condition that needs to be satisfied for the solution to be valid. Overall, the conversation ends with the solution being found and gratitude expressed towards the helpful responses.
Lindsayyyy
Hi,

I was about solving a refrection problem and I'm just one step away from the solution. I stuck at some simple problem I guess.

I have:

$$\alpha=60°+\frac1 3 arcsin(\frac1 {1,33} sin(\alpha))$$

and I want to find the alpha but I have troubles to solve this.

I guess it's not that hard, but I'm failing right now

Hi Lindsayyyy!

The problem you have here has afaik no algebraic solution.
If that is what you're looking for, you won't find it.

What you can do, of course, is make an approximation.
That's easy with for instance http://wolframalpha.com.

Cheers!

Hhhmm...I gave it a try, and I solved by hand pretty easily.

This was my approach:

Multiply both side of the equation by 3, to eliminate the 3 in the denominator.
Move the 180 from the righ to the left
take the sine of both sides of the equation
you should have

sin(3a - 180) = sin( arcsin( (1/1.33) sin(a) ) )

but the sin(p-180) = -sin(p)

-sin(3a) = sin( arcsin( (1/1.33) sin(a) ) )

and the sin ( arcsin(x) ) = x

-sin(3a) = (1/1.33) sin(a)

and sin(3a) = 3sin(a) - 4(sin(a))^3

-3sin(a) + 4(sin(a))^3 = (1/1.33)sin(a)

put everything on one side of the equation and you have a cubic equation without a quadratic or constant term:

4(sin(a))^3 - (3 + 1/1.33)sin(a) = 0

Now, we simply say

X = sin(a)
A = 4
C = -(3 + 1/1.33)

and we have

AX^3 + CX = 0

or, if we factor one X out:

X ( AX^2 + C ) = 0

So, one solution is X1 = zero, due to the single X...now, we need to solve the roots of the quadratic equation with no B term:

X2 = +sqrt ( -C/A ) = 0.9685
X3 = -sqrt ( -C/A ) = -0.9685

So, from X = sin(a)

a = arcsin(X)
a1 = arcsin(0) = 0 degrees
a2 = arcsin(0.9685) = 75.578 degrees
a3 = arcsin(-0.9685) = -75.578 degrees

and that's it.

You still need to check if the solutions work.

You created an extra solution when you replaced

3a - 180 = arcsin( (1/1.33) sin(a) ) with

sin(3a - 180) = sin(arcsin( (1/1.33) sin(a) ))

gsal said:
Hhhmm...I gave it a try, and I solved by hand pretty easily.

Awww! You're right.
I made a mistake in my calculation and came out with sine's with different arguments that could not be made equal.
Good catch!

willem2 said:
You created an extra solution when you replaced ...

Yes. That is right too...

There is an extra condition that -90 < 3a - 180 ≤ 90.
That is, 30 < a ≤ 90.Cheers!

Thank you very much for help. Nice approach @ gsal, this helped me a lot.

## What is the trigonometry reflection problem?

The trigonometry reflection problem involves finding the coordinates of a point after it has been reflected across a line or a plane. This problem is commonly encountered in geometry and is an important concept in trigonometry.

## What are the different types of reflections in trigonometry?

There are three types of reflections in trigonometry: reflection across the x-axis, reflection across the y-axis, and reflection across the origin. Each type of reflection has its own specific rules and formulas for finding the coordinates of the reflected point.

## How do you solve a trigonometry reflection problem?

To solve a trigonometry reflection problem, you will need to know the coordinates of the original point, the type of reflection that is being performed, and the line or plane of reflection. You can then use the appropriate formula to calculate the coordinates of the reflected point.

## What are the applications of the trigonometry reflection problem?

The trigonometry reflection problem has many applications in real-life situations. For example, it can be used in architecture to determine the placement of mirrors or windows in a building. It is also used in computer graphics to create reflections in virtual environments.

## What are some tips for solving trigonometry reflection problems?

Some tips for solving trigonometry reflection problems include carefully identifying the type of reflection, visualizing the reflection in your mind, and using the correct formulas and equations. It is also important to double check your calculations and use a graphing tool to verify your answers.

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