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Chapter 5: Problem 3

Show that the point is on the unit circle. $$\left(\frac{7}{25}, \frac{24}{25}\right)$$

Short Answer

Expert verified
Yes, the point \(\left(\frac{7}{25}, \frac{24}{25}\right)\) is on the unit circle as it satisfies \(x^2 + y^2 = 1\).

Step by step solution

01

Understanding the Unit Circle

The unit circle has a radius of 1 and is centered at the origin (0,0). A point \((x, y)\) is on the unit circle if the equation \(x^2 + y^2 = 1\) is satisfied.
02

Substitute the Point's Coordinates

Substitute \(x = \frac{7}{25}\) and \(y = \frac{24}{25}\) into the equation \(x^2 + y^2 = 1\). This gives \(\left(\frac{7}{25}\right)^2 + \left(\frac{24}{25}\right)^2 = 1\).
03

Calculate Each Square

Calculate \(\left(\frac{7}{25}\right)^2 = \frac{49}{625}\) and \(\left(\frac{24}{25}\right)^2 = \frac{576}{625}\).
04

Add the Squares

Add the results from the previous step: \(\frac{49}{625} + \frac{576}{625} = \frac{625}{625}\).
05

Verify the Equation

Since \(\frac{625}{625} = 1\), the equation \(x^2 + y^2 = 1\) holds true, confirming that the point \(\left(\frac{7}{25}, \frac{24}{25}\right)\) is on the unit circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coordinate Geometry
Coordinate geometry is a branch of geometry where points are defined on a plane using their coordinates鈥攅ssentially their positions on the x-axis and y-axis. A point is represented as \((x, y)\).
  • Origin: The point \((0, 0)\) is called the origin and is the center of the coordinate plane.
  • X-axis: The horizontal axis on the coordinate plane.
  • Y-axis: The vertical axis on the coordinate plane.
Understanding coordinate geometry is fundamental in solving problems involving figures on a plane, such as determining a point's position or whether a point lies on a specific curve like the unit circle.
Equation of a Circle
The equation of a circle in coordinate geometry represents all the points that satisfy the relation around a fixed point, called the center, with a specific distance called the radius.
  • General form: The equation of a circle is usually written as \((x - h)^2 + (y - k)^2 = r^2\), where
    • \((h, k)\) is the center of the circle.
    • \(r\) is the radius.
  • Unit circle: A special type of circle with center \((0, 0)\) and radius 1.
In our example, if a point satisfies the equation \(x^2 + y^2 = 1\) (the equation of a unit circle), then the point lies on the unit circle. This verifies that\(\left(\frac{7}{25}, \frac{24}{25}\right)\) is indeed on the circle.
Radius of a Circle
The radius of a circle is defined as the distance from the center of the circle to any point on its circumference. It is a constant number for a given circle.
  • Unit circle radius: For the unit circle, this distance is always 1.
  • The radius plays a crucial role in the equation of the circle, as it is squared to become the right-hand side of the circle's equation \((x - h)^2 + (y - k)^2 = r^2\).
  • Changes in the radius alter the size of the circle without changing its shape.
Knowing the radius helps find the equation of a circle and verify whether specific points lie within or outside the boundary of this circle. In the case of the unit circle, any point like \(\left(\frac{7}{25}, \frac{24}{25}\right)\) that satisfies \(x^2 + y^2 = 1\) confirms its position exactly on the circle.

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Most popular questions from this chapter

The terminal point \(P(x, y)\) determined by a real number \(t\) is given. Find \(\sin t, \cos t,\) and \(\tan t\) $$\left(-\frac{1}{3},-\frac{2 \sqrt{2}}{3}\right)$$

Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\). $$t=-\frac{41 \pi}{4}$$

Find the values of the trigonometric functions of \(t\) information. $$\tan t=-\frac{3}{4}, \quad \cos t>0$$

Find the values of the trigonometric functions of \(t\) information. \(\sin t=\frac{3}{5}, \quad\) terminal point of \(t\) is in quadrant II

A strong gust of wind strikes a tall building, causing it to sway back and forth in damped harmonic motion. The frequency of the oscillation is 0.5 cycle per second and the damping constant is \(c=0.9 .\) Find an equation that describes the motion of the building. (Assume \(k=1\) and take \(t=0\) to be the instant when the gust of wind strikes the building.)
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