Problem 11 - Precalculus: A Concise Course

Question

Find the point \((x, y)\) on the unit circle that corresponds to the real number \(t\). $$ t=\frac{\pi}{4} $$

Step-by-Step Solution

Answer

The coordinates of the point on the unit circle corresponding to \(t=\frac{\pi}{4}\) are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Step 1: Understand the question and identify known values

We know that the real number \(t=\frac{\pi}{4}\). Also recall that the coordinates of any point on a unit circle can be represented using cosine and sine values of the angle which the point makes with the positive x-axis.

Step 2: Compute the cosine value of the given angle

The x-coordinate of the point on the unit circle is given by \(\cos(t)\), fill in the angle \(t=\frac{\pi}{4}\) to get \(\cos(\frac{\pi}{4})\). Computed, this equals \(\frac{1}{\sqrt{2}}\), or, written alternatively as \(\frac{\sqrt{2}}{2}\).

Step 3: Compute the sine value of the given angle

The y-coordinate of the point on the unit circle is given by \(\sin(t)\), fill in the angle \(t=\frac{\pi}{4}\) to get \(\sin(\frac{\pi}{4})\). Computed, this equals \(\frac{1}{\sqrt{2}}\), or, written alternatively as \(\frac{\sqrt{2}}{2}\).

Step 4: Formulate the point

Thus, the coordinates of the point on the unit circle corresponding to \(t=\frac{\pi}{4}\) are \((\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})\).

Key Concepts

Trigonometry and the Unit Circle

Trigonometry is a branch of mathematics that delves into the relationships between the angles and sides of triangles, specifically right-angled triangles. But beyond triangles, one of its fundamental constructs is the unit circle, which is a circle with a radius of 1 centered at the origin \( (0, 0) \) of a coordinate plane.

The unit circle is incredibly useful because it provides a geometric representation of the trigonometric functions sine and cosine. As you move around the circle, every point on its circumference can be described by two coordinates, which are the values of the sine and cosine functions for a specific angle. These angles are measured in radians \( (2\pi \text{ radians} = 360^\circ) \) and are often referenced from the positive x-axis. For example, an angle of \( \frac{\pi}{4} \) radians corresponds to an angle of 45 degrees.

Importantly, for any angle \( t \) in a unit circle, the coordinates \( (\cos(t), \sin(t)) \) will give you the exact point on the circle's edge. This ties back to the exercise's objective of finding a point on the unit circle using trigonometric functions and the principles of trigonometry.

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that repeat every \( 2\pi \) radians, known as their period. These functions project the motion on the unit circle onto the y-axis (sine) and x-axis (cosine), respectively. When you're given an angle \( t \) like \( \frac{\pi}{4} \) radians, you can find the corresponding point on the unit circle by calculating \( \sin(t) \) for the y-coordinate and \( \cos(t) \) for the x-coordinate.

Both functions are crucial for converting between angular motion and linear motion, making them incredibly valuable in many fields, such as physics, engineering, and even computer graphics.

Interpreting Sine and Cosine Values

When \( t \) is \( \frac{\pi}{4} \), both \( \cos(t) \) and \( \sin(t) \) yield \( \frac{\sqrt{2}}{2} \), which places the point squarely in the first quadrant, where both x and y values are positive. This reflects the angle's position at 45 degrees, equally distant from both the x and y-axes. It demonstrates the symmetry of these functions within the unit circle, particularly for angles in the first quadrant.

Radians: A Measure of Angle

Radians provide a way of measuring angles based on the radius of a circle. One radian is the angle created when the arc opposite that angle is equal to the radius of the circle. Therefore, it links linear distance traveled around a circle's edge directly with the angle measurement.

The exercise involves converting the angle measurement from radians to interpret it on the unit circle. The angle of \( \frac{\pi}{4} \) radians is a commonly used angle in trigonometry because it's precisely one-eighth of the full circle, \( 2\pi \) radians.

Understanding Radians in the Unit Circle

Measuring angles in radians is preferable in mathematics because it simplifies many formulae and calculations. When using radians, it's easier to apply trigonometric functions, since these functions were inherently defined in terms of the unit circle. To understand these concepts more thoroughly, it's beneficial to visualize the unit circle, mark the angle in radians, and then locate the corresponding point using the sine and cosine functions.