Problem 56 - Algebra and Trigonometry

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

In Quadrant IV, \(\sin t = -\sqrt{1 - \cos^2 t}\).

Step 1: Understand the Relationship Between Sine and Cosine

Recognize that in any quadrant, cosine and sine are related through the Pythagorean identity: \( \cos^2 t + \sin^2 t = 1 \). This relationship will be imperative in expressing \( \sin t \) in terms of \( \cos t \).

Step 2: Determine Sign of Functions in Quadrant IV

In Quadrant IV, \( \cos t \) is positive and \( \sin t \) is negative. Understanding the signs helps in establishing the correct form of the trigonometric equations.

Step 3: Isolate \( \sin t \) Using the Pythagorean Identity

We can express \( \sin t \) in terms of \( \cos t \) using the identity \( \sin^2 t = 1 - \cos^2 t \). This yields \( \sin t = \pm \sqrt{1 - \cos^2 t} \).

Step 4: Apply Sign Information from Quadrant IV

Based on Step 2, since \( \sin t \) is negative in Quadrant IV, we select the negative root: \( \sin t = -\sqrt{1 - \cos^2 t} \).

Key Concepts

Pythagorean Identity

The Pythagorean identity is a fundamental concept in trigonometry. It is derived from the Pythagorean theorem, which relates to right triangles in geometry. This identity is expressed as:

• \( \cos^2 t + \sin^2 t = 1 \)

This equation shows how the squares of the cosine and sine of an angle add up to 1. It provides a powerful tool to shift between these two trigonometric functions. Whenever you know the value of one function, the identity allows you to easily calculate the other. For instance, if you know \( \cos t \), you can find \( \sin t \) using:

• \( \sin^2 t = 1 - \cos^2 t \)

By taking the square root, you can solve for \( \sin t \), although you'll need to consider the correct sign (positive or negative) based on the triangle or quadrant involved.

Sine and Cosine Relationship

The sine and cosine functions are closely linked, and their relationship forms the backbone of many trigonometric concepts. These functions represent the ratios of the sides of a right triangle with respect to one of its angles.

The cosine of an angle \( t \) can be thought of as the x-coordinate, while the sine is the y-coordinate of a point on the unit circle centered at the origin. This relationship is captured perfectly in the Pythagorean identity shared earlier.

In practical terms, when you know \( \cos t \), you can determine \( \sin t \) by manipulating the Pythagorean identity. Remembering that both functions have graphs shaped like sine waves, they periodically rise and fall, symbolizing different values depending on the angle \( t \). Understanding this helps in recognizing patterns when solving trigonometric equations.

Quadrant Signs

Recognizing which quadrant an angle terminates in is crucial for determining the signs of its trigonometric functions. The four quadrants of the Cartesian plane affect the positivity or negativity of \( \sin t \) and \( \cos t \).

• Quadrant I: \( \cos t > 0 \), \( \sin t > 0 \)
• Quadrant II: \( \cos t < 0 \), \( \sin t > 0 \)
• Quadrant III: \( \cos t < 0 \), \( \sin t < 0 \)
• Quadrant IV: \( \cos t > 0 \), \( \sin t < 0 \)

When using trigonometric identities like the Pythagorean identity, it's important to apply the correct sign based on the quadrant. For instance, in Quadrant IV, while \( \cos t \) would be positive, \( \sin t \) would be negative. This information dictates whether you choose the positive or negative square root when solving for \( \sin t \). Understanding quadrant signs allows for accurate calculations and helps prevent errors in trigonometric problem-solving.