Problem 2 - Algebra and Trigonometry

Question

If \(P(x, y)\) is on the unit circle, then \(x^{2}+y^{2}=\) _________ . So for all \(t\) we have \(\sin ^{2} t+\cos ^{2} t=\) _______.

Step-by-Step Solution

Answer

The answer to both blanks is 1.

Step 1: Understanding the Unit Circle

The unit circle is defined as a circle with a radius of 1, centered at the origin of the coordinate plane. This means that any point \(P(x, y)\) lying on the unit circle satisfies the equation \(x^2 + y^2 = r^2\). Since the radius \(r\) is 1, the equation simplifies to \(x^2 + y^2 = 1\).

Step 2: Recognizing the Relationship with Trigonometric Functions

On the unit circle, if \(t\) is the angle measured from the positive x-axis to the line connecting the origin to the point \(P\), then the coordinates \(x\) and \(y\) can be represented as \(\cos t\) and \(\sin t\) respectively. Therefore, the equation \(x^2 + y^2 = 1\) translates to \((\cos t)^2 + (\sin t)^2 = 1\).

Step 3: Using Pythagorean Identity

From the above steps, we recognize that \((\cos t)^2 + (\sin t)^2 = 1\) is a fundamental trigonometric identity known as the Pythagorean identity. It is true for all real numbers \(t\). Thus, for every angle \(t\), we have \(\sin^2 t + \cos^2 t = 1\).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental concept in trigonometry that arises from the geometry of a unit circle.
Imagine a circle with a radius of 1, perfectly centered at the origin of a coordinate system. Any point on this circle is described by coordinates \(x, y\). According to the Pythagorean theorem, the sum of the squares of the coordinates of points lying on this circle will always equal the square of the radius.
Since the unit circle has a radius of 1, it simplifies the equation to \(x^2 + y^2 = 1\).

This relation isn't just a geometric curiosity; it underpins the identity \((\sin t)^2 + (\cos t)^2 = 1\), where \(t\) represents an angle. This equation holds true for all angles, acting as a bridge within trigonometric calculations and ensuring accuracy and balance in mathematical equations involving sine and cosine functions. It helps us quickly verify or derive information about angles and trigonometric values.

trigonometric functions

Trigonometric functions connect angles within a unit circle to coordinate values, essentially describing the circular motion and periodic behavior. The primary trigonometric functions include sine, cosine, and tangent, among others. However, sine and cosine play a crucial role when examining the unit circle.

Think of an angle \(t\) formed from the positive x-axis to a line drawn to a point \(P(x, y)\) on the unit circle. The x-coordinate of the point represents the cosine of the angle, and the y-coordinate is the sine. Therefore, for angle \(t\), \((\cos t, \sin t)\) ensures a coordinate representation within the circle.

• Cosine \((\cos t)\) calculates the horizontal distance from the origin to the point P.
• Sine \((\sin t)\) calculates the vertical distance from the origin to the point P.

These functions are powerful, allowing us to decipher various characteristics of wave forms, circular paths, and oscillatory phenomena, governed by the geometry of the unit circle.

cosine and sine

The trigonometric functions cosine and sine have a deep relationship with the unit circle, as they give us a straightforward way to determine coordinates. When you plot a point on a unit circle dictated by an angle \(t\), the cosine and sine provide key insights:

• Cosine measures how far right or left (x-axis) that point is from the center of the circle.
• Sine gauges how far up or down (y-axis) the point sits relative to the center.

They are essential for defining how angles translate into movements and positions on a circle.

The names cosine and sine stem from their mathematical background in right triangles but are extended to work dynamically through the circular scope of angles beyond 0° to 90°, up to 360° and even beyond, covering all four quadrants of the circle. This cyclical nature makes them apt for modeling periodic phenomena like waves and harmonic motion, and their values are defined by the simple yet powerful relationships expressed by the unit circle, as encapsulated by the Pythagorean identity.