Problem 55 - Precalculus Mathematics for Calculus

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\sin t, \cos t ; \quad\) Quadrant II

Step-by-Step Solution

Answer

\(\cos t = -\sqrt{1 - \sin^2 t}\) in Quadrant II.

Step 1: Understanding Quadrants

The unit circle is divided into four quadrants. In Quadrant II, the sine value is positive, and the cosine value is negative.

Step 2: Expressing Sine in terms of Cosine

In general, for any angle on the unit circle, the identity \( an t = \frac{\sin t}{\cos t}\) relates sine and cosine. However, we aren't given tangent or need it directly, just noting that sine is positive and cosine is negative here.

Step 3: Using the Pythagorean Identity

Recall the Pythagorean Identity: \(\sin^2 t + \cos^2 t = 1\). We manipulate this to express one in terms of the other. Solving for \(\cos t\), we get \(\cos t = -\sqrt{1 - \sin^2 t}\) in Quadrant II, because cosine is negative there.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that helps us relate angles to the coordinates on the unit circle. It specifically corresponds to the y-coordinate of a point on the circle. When you think of the sine function, imagine a circle with a radius of 1 unit. The sine of an angle \(t\) (often denoted as \(\sin t\)) represents the height of that point above the x-axis. This is particularly easy to visualize when dealing with angles positioned on the unit circle.

• The sine function ranges between -1 and 1, because the highest or lowest point a y-coordinate can reach on a unit circle is 1 or -1.
• In Quadrant II of the unit circle, sine values are positive, meaning that the points are above the x-axis.
• Useful identity: \(\sin^2 t + \cos^2 t = 1\), which is a cornerstone of trigonometry called the Pythagorean Identity.

Understanding these properties will help you manipulate and understand expressions involving the sine function, especially with respect to other trigonometric functions.

Cosine Function

The cosine function is another core component of trigonometry, crucial for mapping angles to the coordinates on the unit circle. Where sine gives us the vertical position, the cosine function provides the horizontal position. Specifically, \(\cos t\) represents the x-coordinate of the point on the unit circle.

• Just like sine, the cosine function ranges between -1 and 1, corresponding to the width of the unit circle.
• In Quadrant II, where our current problem is set, cosine values are negative, indicating that the x-coordinate falls to the left of the y-axis.
• The identity \(\sin^2 t + \cos^2 t = 1\) also applies, allowing one function to be expressed in terms of the other, as done in the solution step: \(\cos t = -\sqrt{1 - \sin^2 t}\).

Recognizing these basics and the signs in each quadrant helps create connections between trigonometric identities and graphical representations.

Unit Circle

The unit circle is a visual and mathematical tool used to better understand trigonometric functions. It is essentially a circle centered at the origin of a coordinate plane with a radius of 1.

• The circle facilitates understanding the periodic nature of trigonometric functions like sine and cosine.
• Each point on the unit circle can be represented by interchangeable coordinates \((\cos t, \sin t)\). These help relate an angle \(t\) to particular trigonometric values via their association with the x and y axes.
• In Quadrant II, angles are typically between 90° and 180°. While the sine values are positive, meaning above the x-axis, cosine values are negative, meaning to the left of the y-axis.

The unit circle serves as a grounding tool for connecting angles and trigonometric functions, supporting better visualization and manipulation of complex trigonometric expressions.