What does this equation mean?

[inuka00123]

Homework Statement

solove tanα = X/Y

Relevant Equations

tanα = X/Y

does this mean to find tanα you need to divide X/Y or divide (X/Y)/tan to find α?

 

[Gavran]

to find tanα you need to divide X/Y

 

[Mark44]

Homework Statement: solove tanα = X/Y
Relevant Equations: tanα = X/Y

does this mean to find tanα you need to divide X/Y or divide (X/Y)/tan to find α?

First, for a problem asking you to solve an equation with several variables, the problem will usually ask you to solve for one of the variables.
Second, this equation looks to me like an identity, a type of equation that is always true for any combination of the variables in the equation. An example of an identity is $2(x + y) = 2x + 2y$.
Third, the expression on the left side of your equation, $\tan \alpha$, does NOT mean $tan$ times $\alpha$, so it makes no sense at all to divide by $tan$ to get $\alpha$ by itself.

 

[FactChecker]

You should be careful. That equation should be more clearly written as $\tan(\alpha) = X/Y$.
$\tan(\alpha)$ is the tangent function of $\alpha$. After dividing $X/Y$ to get some number (like 0.57735), you need to find out what value of $\alpha$ gives $\tan(\alpha)=0.57735$. The tangent function is complicated to calculate. In general, you should use tables or a calculator to find the "arctan" (inverse function of tangent) of 0.57735.

 

[Mark44]

Second, this equation looks to me like an identity, a type of equation that is always true for any combination of the variables in the equation.

I forgot to add that in a right triangle with legs of length X and Y and an angle α opposite to side X, then tan⁡(α)=XY. For example, in a right triangle whose legs are each 1 unit, with α being one of the acute angles, then tan⁡(α)=11=1. The arctangent function can be used to determine that α=π4 or 45∘.

 

[pasmith]

In general, you should use tables or a calculator to find the "arctan" (inverse function of tangent).

Most calculators will label it as "tan^-1" rather than "arctan".
In mathematics we generally measure angles in radians rather than degrees, so make sure that the calculator is set to the right unit.

 

[chwala]

Most calculators will label it as "tan^-1" rather than "arctan".
In mathematics we generally measure angles in radians rather than degrees, so make sure that the calculator is set to the right unit.

Could you clarify when we should use the notation 'tan⁻¹' versus 'arctan' in mathematics? Also, how do we determine when to work in degrees versus radians — are there specific contexts where one is more appropriate than the other, or does it depend on personal or regional preference?"

 

[phinds]

Could you clarify when we should use the notation 'tan⁻¹' versus 'arctan' in mathematics?

They are inverse functions. That's like asking, when should we use X vs 1/X. It depends entirely on the context.

See post #11

 

[phinds]

how do we determine when to work in degrees versus radians

Google is your friend.

 

[Gavran]

They are inverse functions. That's like asking, when should we use X vs 1/X. It depends entirely on the context.

I suppose you mean $X^{-1}$ vs $1/X$.

 

[phinds]

I suppose you mean $X^{-1}$ vs $1/X$.

HA. Thanks. I missed the -1 and thought he was asking about tan vs arctan. My bad.

 

[TensorCalculus]

Could you clarify when we should use the notation 'tan⁻¹' versus 'arctan' in mathematics? Also, how do we determine when to work in degrees versus radians — are there specific contexts where one is more appropriate than the other, or does it depend on personal or regional preference?"

Somewhat depends.

For tan-1 vs arctan, both are used pretty interchangeably. From personal experience most calculators as well as more modern resources use tan-1. More older books tend to use arctan. It honestly comes down to personal preference- arctan may be used just because its easier to type out or format. But tan-1 is often the first type to be taught in middle or high school, at least in my experience.

For degrees vs radians it's a bit more complex. When you're measuring angles in day to day things, you would use degrees most of the time. You won't see protractors with radians on them or anything, because there are 2pi radians in a circle and the numbers on the protractor would be absurd decimals. And when you calculate things to do with these measured angles, you use degrees. For most mechanics problems you use degrees because it's easier to say 30 degrees than 0.524(etc etc) radians. In chemistry you use degrees to talk about the angles between chemical bonds, and you use degrees to talk about the angles in a triangle or pentagon or hexagon.
But radians are useful too. Especially when you're looking at things in terms of oscillations or rotational movement over time. Or when you're looking at circles. It turns out that when you use radians, a lot of very simple, beautiful relationships come to life. For example, the arclength of a segment on a circle is the radius times the angle the circle subtends, but only if the angle is in radians. The calculus of trig functions is simplest when the angles are in radians. When you look at waves and oscillations, radians come up yet again in the form of angular frequency and phase. Other relationships only work with radians too, like the Rayleigh criterion equation which says that the minimum resolvable angle for a telescope (or more precisely diffraction limited instrument with a circular lens/mirror) is 1.22 times the wavelength of the light coming in divided by the diameter of the lens/mirror. This only works if theta is in radians!

So for degrees vs radians, it's all about which one is more useful for the task. Degrees are useful when you want nice, clean numbers but radians open up a whole world of mathematical relationships that are often very simple and very elegant.
Hope that helps!

 

[votingmachine]

I suppose they might be asking you to go from:
tan = sin / cos to x / y

Then you would use a right triangle with sides x and y and hypotenuse z and the definitions of sin and cosine.

But it seems a bit definitional to me.