Problem 12 - Precalculus with Calculus Previews

Question

Find the exact value of (a) \(\sin t\) and (b) \(\cos t\) for the given value of \(t\). Do not use a calculator. $$ t=3 \pi $$

Step-by-Step Solution

Answer

\( \sin 3\pi = 0 \); \( \cos 3\pi = -1 \).

Step 1: Understand the Problem

We are given that \( t = 3\pi \) and are tasked with finding the exact values of \( \sin t \) and \( \cos t \). Since this angle is a multiple of \( \pi \), we will analyze it using the unit circle.

Step 2: Identify Position of \( t \) on the Unit Circle

In the unit circle, an angle \( 2\pi \) corresponds to a full rotation, landing back at the starting point \((1, 0)\). The angle \( t = 3\pi \) can be decomposed as \( 2\pi + \pi \), which means a full rotation plus half a rotation, ending at the point \((-1, 0)\). This is on the negative x-axis.

Step 3: Determine \( \sin t \)

The \( y \)-coordinate of the point on the unit circle gives us \( \sin t \). For \( t = 3\pi \), the point is \((-1, 0)\), so \( \sin 3\pi = 0 \).

Step 4: Determine \( \cos t \)

The \( x \)-coordinate of the point on the unit circle provides \( \cos t \). For \( t = 3\pi \), the coordinate is \( (-1, 0) \), so \( \cos 3\pi = -1 \).

Key Concepts

Unit Circle

The unit circle is an essential concept in trigonometry that helps to understand trigonometric functions. It is a circle with a radius of 1, centered at the origin of a coordinate system. This means any point \(x, y\) on the circle satisfies the equation \(x^2 + y^2 = 1\). The unit circle is crucial because it provides a geometric representation of angles and their corresponding trigonometric values.

• One complete revolution around the circle is \(2\pi\) radians.
• The circle enables you to visualize angles and their trigonometric ratios like sine and cosine.
• Each angle in standard position on the unit circle is measured from the positive x-axis.

When an angle \(t\) is given, it corresponds to a specific point on the unit circle. The coordinates of that point give the sine and cosine values for that angle. In this exercise, \(t = 3\pi\) means you make one and a half complete revolutions around the circle, ending up at the negative x-axis, specifically at the point \((-1, 0)\).

Sine Function

The sine function, represented as \(\sin\), shows the relationship between an angle in the unit circle and the y-coordinate of its corresponding point. For any given angle \(t\), \(\sin t\) is the y-value of the point where the terminal side intersects the unit circle.

• Sine measures how far up or down a point on the unit circle is from the x-axis.
• For example, \(\sin 0 = 0\), corresponding to the point \(1, 0\) on the unit circle.

In our exercise, \(t = 3\pi\) positions us at \((-1, 0)\). Here, the y-coordinate is 0, which means \(\sin 3\pi = 0\). Sine values repeat every \(2\pi\), so patterns can be predicted for other angles.

Cosine Function

The cosine function, denoted as \(\cos\), is linked to the x-coordinate of a point on the unit circle for a given angle \(t\). \(\cos t\) provides the horizontal distance of that point from the center of the circle.

• Cosine depicts how far left or right a point is along the x-axis.
• For instance, \(\cos 0 = 1\), since the point is at \(1,0\).

In this case, with \(t = 3\pi\), you rotate to \((-1, 0)\). The x-coordinate here is -1, so \(\cos 3\pi = -1\). Cosine also has a periodicity of \(2\pi\), which means it returns to its starting value after every full rotation. This makes it predictable for calculating trigonometric values of angles.