Problem 53 - 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. \(\sin t, \cos t ; \quad\) quadrant II

Step-by-Step Solution

Answer

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

Step 1: Analyze the Given Quadrant

In quadrant II, the sine function is positive, and the cosine function is negative. This information will guide the transformations done in further steps.

Step 2: Use the Pythagorean Identity

Recall the Pythagorean identity: \(\sin^2 t + \cos^2 t = 1\). This can be used to express the cosine function in terms of the sine function.

Step 3: Isolate \(\cos t\)

Solve the Pythagorean identity for \(\cos t\): \[\cos^2 t = 1 - \sin^2 t\].

Step 4: Apply the Sign of Cosine

Since \(\cos t\) is negative in quadrant II, take the negative square root: \(\cos t = -\sqrt{1 - \sin^2 t}\).

Step 5: Express \(\sin t\) in Terms of \(\cos t\)

Using the derived relationship \(\cos t = -\sqrt{1 - \sin^2 t}\), substitute into the original expression to express \(\sin t\) in terms of \(\cos t\): \[\sin t = \sqrt{1 - \cos^2 t}\].

Key Concepts

Pythagorean Identity

When diving into trigonometry, one of the fundamental concepts to grasp is the Pythagorean identity. This identity is a cornerstone in understanding relationships between trigonometric functions. It is expressed in the formula: \[ \sin^2 t + \cos^2 t = 1 \]Here, the terms \( \sin^2 t \) and \( \cos^2 t \) are the squares of the sine and cosine functions respectively. This identity is derived from the Pythagorean theorem and applies to any angle \( t \). It serves as a handy tool to express one trigonometric function in terms of another.
This is especially useful when analyzing how these functions behave in different quadrants of the coordinate plane. It allows us to solve for one function knowing the value of the other by rearranging the terms:

• If you know \( \sin^2 t \), and need \( \cos^2 t \), you simply compute \( 1 - \sin^2 t \).
• Conversely, if you know \( \cos^2 t \), to find \( \sin^2 t \), calculate \( 1 - \cos^2 t \).

This relationship is vital for solving and simplifying trigonometric expressions.

Quadrant Analysis

In trigonometry, understanding how angles and their corresponding trigonometric functions behave in different quadrants of the unit circle is crucial. The coordinate plane is divided into four quadrants, and each has unique characteristics influencing the signs of the sine and cosine functions. Quadrants can be analyzed considering their specific properties:

• In Quadrant I, both sine and cosine are positive.
• In Quadrant II, sine remains positive, but cosine turns negative.
• Moving to Quadrant III, both sine and cosine are negative.
• Finally, in Quadrant IV, sine is negative, and cosine becomes positive.

For our problem, we're particularly interested in Quadrant II. When converting one trigonometric function in terms of another, quadrant analysis helps determine the correct sign of your solution. In this specific case, since sine is positive and cosine is negative in Quadrant II, the derived cosine from the Pythagorean identity will take on a negative value.

Sine Function

The sine function is a fundamental trigonometric function that describes the y-coordinate of a point on the unit circle corresponding to a given angle \( t \). When learning about trigonometry, students often start with the sine function as it directly represents the vertical component of an angle in standard position.Some key properties of the sine function include:

• It oscillates between -1 and 1.
• It is an odd function, thus \( \sin(-t) = -\sin(t) \).
• Sine is positive in Quadrants I and II (where our exercise is focused).

From the Pythagorean identity, we know that to express \( \sin t \) in terms of \( \cos t \) when \( \sin^2 t + \cos^2 t = 1 \), we can rearrange and compute:\[ \sin t = \sqrt{1 - \cos^2 t} \]This formula will yield a positive value since we are in Quadrant II, validating the quadrant analysis. It helps in finding the sine value when you already know the cosine value, highlighting the utility of this key relationship.

Cosine Function

The cosine function, closely related to the sine function, represents the x-coordinate of a point on the unit circle for a given angle \( t \). Just like the sine function, understanding cosine is foundational in trigonometry because it describes the horizontal component in a trigonometric context.Here are some important characteristics of the cosine function:

• It oscillates between -1 and 1, similar to the sine function.
• It is an even function, meaning \( \cos(-t) = \cos(t) \).
• Positive in Quadrants I and IV, and negative in Quadrants II and III.

In terms of our exercise situated in Quadrant II, if we know the sine value, we can use the Pythagorean identity to determine the cosine value. Given that cosine is negative in this quadrant, applying the identity would result in:\[ \cos t = -\sqrt{1 - \sin^2 t} \]This expression ensures the sign is accurately represented for Quadrant II conditions. Understanding how cosine behaves across different quadrants is crucial for correctly solving trigonometric identities and transformations.