Problem 109 - Precalculus

Question

If \(\tan \theta=\frac{a}{b},\) where \(a\) and \(b\) are positive, and if \(\theta\) lies in quadrant III, find \(\sin \theta\)

Step-by-Step Solution

Answer

\(\sin \theta = -\frac{a}{\sqrt{a^2 + b^2}}\)

Step 1: Understand the problem

We are given that \(\tan \theta = \frac{a}{b}\) and that \(\theta\) is in the third quadrant, where both sine and cosine are negative. We need to find \(\sin \theta\).

Step 2: Relate tangent to sine and cosine

Recall that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\). Since \(\tan \theta = \frac{a}{b}\), it implies \(\sin \theta = a \) and \(\cos \theta = b\) with consideration of their signs in quadrant III.

Step 3: Calculate the hypotenuse with Pythagorean identity

Use the identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find the hypotenuse \(r\). Calculate it as follows: \(r = \sqrt{a^2 + b^2}\).

Step 4: Apply trigonometric identities

Since sin is negative in quadrant III, use \(\sin \theta= -\frac{a}{\sqrt{a^2 + b^2}}\). The negative sign is because \(\theta\) is in quadrant III where sine is negative.

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions, usually denoted by \( \tan \theta \). It is defined as the ratio of the lengths of the opposite side to the adjacent side in a right-angled triangle. Mathematically, it is expressed as:

• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

This function is vital when dealing with angles and triangles, as it helps in converting between trigonometric functions. In the context of unit circles and angles on a coordinate plane, the tangent of an angle \( \theta \) can also be expressed via the sine and cosine functions:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

In our exercise, knowing \( \tan \theta = \frac{a}{b} \) helps identify the relationship between \( \sin \theta \) and \( \cos \theta \). Understanding this concept allows us to further explore and solve problems involving trigonometric identities.

Sine Function

The sine function, symbolized as \( \sin \theta \), represents the ratio of the length of the side opposite the angle to the hypotenuse in a right triangle:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)

This function is key in determining the vertical component of a point on the unit circle as it moves through different angles. In our given problem, the sine function is directly linked to the tangent function, by virtue of the relationship specified with the cosine function. Because the problem is situated in the third quadrant, where the sine value is negative, the final calculation for \( \sin \theta \) reflects this with its negative value.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are always true for every value of the variables involved. One of the most widely used identities is the Pythagorean identity, which states:

• \( \sin^2 \theta + \cos^2 \theta = 1 \)

This identity is crucial in problems involving trigonometric functions because it relates \( \sin \theta \) and \( \cos \theta \), allowing us to calculate one from the other. In the provided solution, we use this identity to find the hypotenuse \( r \) of the triangle in the context of the unit circle:

• First, find \( r = \sqrt{a^2 + b^2} \) to determine the hypotenuse.
• Use this \( r \) to express \( \sin \theta \) and \( \cos \theta \) in terms of \( a \) and \( b \).

These identities simplify and solve complex trigonometric equations and are valuable tools in the kit of trigonometry.

Quadrant System

The quadrant system divides the coordinate plane into four sections, based on the x and y axes. Measured counterclockwise from the positive x-axis, the quadrants are numbered I through IV:

• Quadrant I: where both x and y are positive.
• Quadrant II: x is negative, and y is positive.
• Quadrant III: both x and y are negative.
• Quadrant IV: x is positive, and y is negative.

Each quadrant has unique properties regarding the signs of trigonometric functions. In the context of our problem:

• The angle \( \theta \) is in Quadrant III, which means both the sine and cosine values are negative.
• This affects the calculation of trigonometric values, specifically as signet in equations or functions.

Understanding the quadrant system is essential for correctly interpreting the signs of trigonometric functions and simplifying complex calculations.