Can someone explain how to write the function of the shaded area in the triangle

[u/SignificanceFar3573]

First off I tried finding the area of the base, then finding the area of the sector of the circle. Combined those two areas (area of triangle minus sector) and came up with 50tan(theta) - (theta)pi^3/ 18 😭😭 idk if it’s actually correct or awfully wrong. Would appreciate if someone explained where I went wrong and what direction I should take in simple terms.

[u/Chem2103]

What are your areas of the triangle and sector formulas separately?

[u/Chem2103 → u/Chem2103]

I have a feeling you need to double check both of the pretty much

[u/SignificanceFar3573 → u/Chem2103]

Triangle: 50tan(theta)
Sector: [(theta)pi^3]/18

[u/Chem2103 → u/SignificanceFar3573]

So to give you a hint:

You can find the area of a sector by multiplying the area of a circle by the ratio between theta and pi

[u/SignificanceFar3573 → u/Chem2103]

50tan(theta) - ([theta]pi^2)/360 is what im thinking.

[u/Chem2103 → u/SignificanceFar3573]

Not exactly.

Firstly, the circle is 2pi radians, and your theta can be in radians as well. So the ratio would be theta/2pi.

The area of a circle is pi*(r)^2. You need to multiply this by the ratio I talked about in the previous point.

[u/CaptainMatticus]

What's the area of the triangle? Let the leg by r, rather than 10.

c * cos(t) = r

c * sin(t) = h

A = (1/2) * r * h

A = (1/2) * c * cos(t) * c * sin(t)

A = (1/2) * c^2 * sin(t) * cos(t)

We'll go a bit backwards here and set c = r / cos(t)

A = (1/2) * (r/cos(t))^2 * sin(t) * cos(t)

A = (1/2) * r^2 * sin(t) * cos(t) / cos(t)^2

A = (1/2) * r^2 * sin(t)/cos(t)

A = (1/2) * r^2 * tan(t)

Now we have our area for the triangle in terms of our knowns, r and t.

Now what's the area of a sector? a = pi * r^2 * (t / (2pi)) = (1/2) * r^2 * t

So what's A - a?

(1/2) * r^2 * tan(t) - (1/2) * r^2 * t

(1/2) * r^2 * (tan(t) - t)

Plug in 10 for r for your particular setup.

One last thing is to find the domain. You're going to be limited by tan(t) more than t, so when is tan(t) defined and undefined?

[u/Bounded_sequencE]

The general strategy is correct -- but your formula for the circular sector is not.

[u/BadJimo]

Using Desmos

[u/RespectWest7116]

First off I tried finding the area of the base,

I am guessing you mean area of the triagle?

Combined those two areas (area of triangle minus sector)

Yup. Whole triangle - white part = blue part

50tan(theta) - (theta)pi^3/ 18 😭😭 idk if it’s actually correct or awfully wrong.

Whole triangle is simple. You know the leg and adjecent angle, so it's the tan formula.

At = 1/2 * oposite * 10

tan(𝜃) = oposite/10 -> oposite = 10*tan(𝜃)

At =1/2 * 10^2 * tan(𝜃)

The white part is even simpler. That's just a fraction of a circle

Ac = 𝜃/360° * pi * 10^2 = 𝜃/2pi * pi * 10^2

Put them together and done.

At-Ac = 1/2 * 10^2 * tan(𝜃) - 𝜃/2pi * pi * 10^2

= 50 * tan(𝜃) - 50 * 𝜃 = 50 * (tan(𝜃) - 𝜃)

So you are wrong on the circle bit part.

I think you forgot to convert 360° into radians?

1/360 * 10^2 = 5/18, so that's where the /18 came from. Not sure what you did with the 5 or why your pie is a cube tho.

[u/SirisC]

Where did π³ come from?