# Solve "Circles and Sectors: 3θ=2(π−sinθ)

• look416
In summary, the problem involves a semicircle with diameter AB and point P on the semicircle such that the area of sector POB is twice the area of the shaded segment. The goal is to show that 3 times the angle POB is equal to 2 times the difference of pi and the sine of angle POB. To solve this, we use the formulas for the area of a circle and the area of a segment. The angle of the shaded segment can be found by subtracting angle POB from pi. From there, we can plug in the known information and solve for the angle POB.
look416

## Homework Statement

The diagram shows a semicircle APB on AB as diameter. The midpoint of AB is O. The point P on the semicircle is such that the area of the sector POB is equal to twice the area of the shade segment. Given that angle POB is $$\theta$$ radians, show that

3$$\theta$$ = 2($$\pi$$-sin$$\theta$$)​

## The Attempt at a Solution

using formula
Area of circle = $$\frac{1}{2}$$r2$$\theta$$
and
Area of segment = $$\frac{1}{2}$$r2 ($$\theta$$ - sin $$\theta$$ )
heres the problems
from the picture http://img130.imageshack.us/img130/1790/001tz.jpg
questions 4
the the angle of the segment is $$\pi$$-$$\theta$$
there I am clueless even i inserted the info i have
what i really get is
$$\theta$$=2[$$\pi$$-$$\theta$$-sin($$\pi$$-$$\theta$$)]​
of course we can't use formula blindly so anyone can help me there

Last edited by a moderator:

If sector POA contains the shaded segment and triangle POA, how do you find the area of the shaded region?

I would approach this problem by first analyzing the given information and equations to fully understand the problem at hand. In this case, we are given a semicircle with a point P on its circumference and we are asked to show that 3θ = 2(π-sinθ). This statement involves both angles and areas, so we need to consider both aspects in our solution.

To start, let's consider the area of the sector POB. This can be expressed as ½r2θ, where r is the radius of the semicircle and θ is the central angle of the sector. We also know that the area of the shaded segment is equal to twice the area of the sector. This can be represented as 2(½r2(θ-sinθ)). So, we can set these two expressions equal to each other:

½r2θ = 2(½r2(θ-sinθ))

Simplifying this equation, we get:

r2θ = r2(θ-sinθ)

Now, let's consider the angle POB. We know that it is equal to θ radians. We also know that the angle of the shaded segment is π-θ radians, as stated in the problem. So, we can write the following equation:

θ + (π-θ) = π

Using the trigonometric identity sin(π-x) = sinx, we can rewrite this equation as:

θ + sinθ = π

Substituting this into our original equation, we get:

r2(θ + sinθ) = r2(θ-sinθ)

Expanding this equation, we get:

r2θ + r2sinθ = r2θ - r2sinθ

Simplifying, we get:

r2sinθ = -r2sinθ

Since the left and right sides of the equation are equal, we can conclude that the equation is true for any value of r and θ. Therefore, we have shown that 3θ = 2(π-sinθ).

## What is the equation for "Solve "Circles and Sectors: 3θ=2(π−sinθ)"?

The equation is 3θ=2(π−sinθ).

## What is the value of θ in "Solve "Circles and Sectors: 3θ=2(π−sinθ)"?

The value of θ is not specified in the equation. It can be any value that satisfies the equation.

## How do I solve "Circles and Sectors: 3θ=2(π−sinθ)" for θ?

To solve this equation, you can use algebraic manipulation techniques to isolate θ on one side of the equation. You can also use a calculator or computer program to find the numerical value of θ that satisfies the equation.

## What is the significance of "Circles and Sectors: 3θ=2(π−sinθ)" in science?

This equation is commonly used in geometry and trigonometry to calculate the area and arc length of a sector of a circle. It is also used in physics and engineering to calculate the motion of objects moving in circular paths or orbits.

## What are some real-life applications of "Circles and Sectors: 3θ=2(π−sinθ)"?

This equation can be applied in various fields such as engineering, architecture, and navigation. For example, it can be used to calculate the area of a circular garden or the distance traveled by a satellite in orbit around the Earth.

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