Problem 55 - 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

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

Step 1: Identifying Trigonometric Identities

In any quadrant, the sine and cosine functions are related by the identity \( \sin^2 t + \cos^2 t = 1 \). This identity will form the basis of our transformation.

Step 2: Understanding Quadrant II Sign Convention

In Quadrant II, the sine function is positive, while the cosine function is negative due to the position relative to the axes. This means \( \sin t > 0 \) and \( \cos t < 0 \).

Step 3: Expressing \(\sin t\) in terms of \(\cos t\)

We start with the Pythagorean identity: \( \sin^2 t + \cos^2 t = 1 \). Solving for \( \sin^2 t \), we get \( \sin^2 t = 1 - \cos^2 t \). Taking the square root, we have \( \sin t = \sqrt{1 - \cos^2 t} \), keeping the positive square root since \( \sin t > 0 \) in Quadrant II.

Step 4: Final Relationship

Thus, the expression for \( \sin t \) in terms of \( \cos t \) in Quadrant II is \( \sin t = \sqrt{1 - \cos^2 t} \).

Key Concepts

Sine and Cosine Functions

The sine and cosine functions are fundamental in trigonometry and are represented by the symbols \(\sin t\) and \(\cos t\) respectively. They are defined using the unit circle, which is a circle with a radius of one, centered at the origin of the coordinate plane.

• Sine Function (\(\sin t\)): It measures the y-coordinate of an angle \(t\) on the unit circle.
• Cosine Function (\(\cos t\)): It measures the x-coordinate of an angle \(t\) on the unit circle.

Together, these functions help in determining the position of an angle's terminal side in a coordinate system. With their periodic nature, they also play a crucial role in modeling wave-like phenomena. Different angles will yield different sine and cosine values, which are key for computing other trigonometric functions and solving geometric problems.

Quadrants in Trigonometry

In trigonometry, the coordinate plane is divided into four quadrants. This helps to categorize angles based on their location, affecting the signs of sine and cosine functions.

• Quadrant I: Both \(\sin t\) and \(\cos t\) are positive.
• Quadrant II: \(\sin t\) is positive and \(\cos t\) is negative.
• Quadrant III: Both \(\sin t\) and \(\cos t\) are negative.
• Quadrant IV: \(\sin t\) is negative and \(\cos t\) is positive.

This classification comes from the signs of the x and y coordinates in each quadrant. The knowledge of quadrants is particularly useful for solving trigonometric identities and equations, like determining the angle related to a given sine or cosine value. In our example problem, the emphasis is on Quadrant II, where understanding the sign conventions is crucial for accurate transformations of trigonometric expressions.

Pythagorean Identity

The Pythagorean identity is one of the most important identities in trigonometry. It connects the sine and cosine functions in a simple but powerful way:\[\sin^2 t + \cos^2 t = 1\]This equation is derived from the equation of a circle, \(x^2 + y^2 = r^2\), by setting the radius \(r = 1\) (the unit circle).

Applications of the Identity

The Pythagorean identity is instrumental in simplifying expressions and solving equations:

• It allows us to express one trigonometric function in terms of another, as seen in the original exercise.
• If you know one value, say \(\cos t\), you can easily find \(\sin t\) using \(\sin^2 t = 1 - \cos^2 t\).

In Quadrant II, where cosine is negative, the identity helps assure that we choose the correct sign for sine based on its positive nature. This identity is fundamental for understanding how trig functions interact and relate on the unit circle.