Problem 56 - 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. \(\cos t, \sin t ; \quad\) Quadrant IV

Step-by-Step Solution

Answer

\(\sin t = -\sqrt{1 - \cos^2 t}\)

Step 1: Understand the Quadrant

In the fourth quadrant of the unit circle, the cosine of any angle is positive, and the sine is negative. This is important for determining the signs of trigonometric functions.

Step 2: Analyze the given expression

We are given the expressions \(\cos t\) and \(\sin t\). We need to express \(\sin t\) in terms of \(\cos t\) considering the quadrant.

Step 3: Use the Pythagorean Identity

The Pythagorean identity is \(\cos^2 t + \sin^2 t = 1\). We can rearrange this to find \(\sin^2 t = 1 - \cos^2 t\).

Step 4: Solve for \(\sin t\) considering the sign

Since \(t\) is in Quadrant IV, \(\sin t\) must be negative. Thus, \(\sin t = -\sqrt{1 - \cos^2 t}\).

Step 5: Write the final expression

The expression for \(\sin t\) in terms of \(\cos t\) in the fourth quadrant is \(\sin t = -\sqrt{1 - \cos^2 t}\).

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental concept in trigonometry that relates the squares of the sine and cosine of an angle. This identity is derived from the Pythagorean Theorem which applies to right-angled triangles. For any angle, the Pythagorean Identity can be expressed as

• \(\cos^2 t + \sin^2 t = 1\)

This equation is extremely useful because it allows us to express one trigonometric function in terms of another. For example, if you know the cosine of an angle, you can find the sine using

• \(\sin^2 t = 1 - \cos^2 t\)
• \(\sin t = \pm\sqrt{1 - \cos^2 t}\)

Understanding the quadrant in which the terminal side of the angle resides helps determine the correct sign of the sine or cosine function.

Unit Circle

The Unit Circle is an essential tool in trigonometry, providing a geometric representation of the various angles and their corresponding trigonometric values. It is a circle with a radius of 1, centered at the origin of the coordinate plane.

• On the unit circle, the coordinates of any point \((x, y)\) are equal to \((\cos t, \sin t)\) where \(t\) is the angle formed with the positive x-axis.
• This unique setup allows us to obtain the sine and cosine values directly from the x and y coordinates of the point on the circle.

The unit circle simplifies the understanding of how trigonometric functions behave as angles increase or decrease, whether in standard position, traveling counterclockwise for positive angles, or clockwise for negative angles.

Angle Quadrants

In trigonometry, the coordinate plane is divided into four quadrants, each having specific sign rules for sine and cosine.

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive while cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Cosine is positive, but sine is negative.

Understanding these quadrants is crucial when applying trigonometric functions. For instance, when solving problems involving angles and the unit circle, knowing which quadrant the angle lies in helps determine the appropriate signs for the trigonometric functions. This knowledge makes it easier to handle expressions like \(\sin t = -\sqrt{1 - \cos^2 t}\) which rely on the quadrant to assign the correct sign.