Problem 29 - Precalculus

Question

Find the indicated trigonometric function values. If \(\tan \theta=\frac{84}{13},\) and the terminal side of \(\theta\) lies in quadrant III, find \(\sin \theta\)

Step-by-Step Solution

Answer

\(\sin \theta = -\frac{84}{85}\) in quadrant III.

Step 1: Recall the Identity for Tangent

The identity for tangent says that \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\). Given \(\tan \theta = \frac{84}{13}\), it suggests one leg of the right triangle has a length of 84 and the other leg has a length of 13.

Step 2: Determine the Hypotenuse of the Triangle using Pythagorean Theorem

Use the Pythagorean theorem to find the hypotenuse. \(c = \sqrt{a^2 + b^2}\) where \(a = 84\) and \(b = 13\). Calculate: \[c = \sqrt{84^2 + 13^2} = \sqrt{7056 + 169} = \sqrt{7225} = 85\]

Step 3: Define Sine in Terms of Opposite and Hypotenuse

The sine of an angle is the ratio of the length of the opposite side to the hypotenuse. Thus, \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{84}{85}\).

Step 4: Apply the Sign of Sine in Quadrant III

Since \(\theta\) is in quadrant III, both sine and cosine are negative. Therefore, \(\sin \theta = -\frac{84}{85}\).

Key Concepts

Pythagorean Theorem

Many students first get introduced to the Pythagorean Theorem in geometry class as a fundamental principle for finding sides of a right triangle. It states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In formula terms, it looks like this:
\[ c^2 = a^2 + b^2 \]
When using the Pythagorean Theorem to solve for a missing side, always make sure you know which side is your hypotenuse. In the problem you are working on, you have a right triangle in which the tangent of angle \( \theta \) has values for the opposite and adjacent sides.
You worked out the hypotenuse as follows:

• Opposite side: \( 84 \)
• Adjacent side: \( 13 \)
• Hypotenuse: \[ c = \sqrt{84^2 + 13^2} = \sqrt{7225} = 85 \]

This formula is useful not only in trigonometry but also in physics and engineering, where relationships between three-dimensional points are common.

Tangent Function

The tangent function is one of the basic trigonometric functions and is primarily used to relate the angles of a right triangle to the lengths of the sides. The tangent of an angle \( \theta \) in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side.
In formula terms, it's expressed as:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
In this exercise,

• \( \tan \theta \) is given as \( \frac{84}{13} \), which means the opposite side is \( 84 \) and the adjacent side is \( 13 \)

Understanding tangent is beneficial for measuring angles and distances in applications like navigation, optics, and architecture.
Remember that the sign of the tangent function depends on the quadrant the angle \( \theta \) is in. Since this problem places \( \theta \) in quadrant III, the tangent value remains positive.

Sine Function

The sine function is another fundamental trigonometric function, often used to calculate various properties of angles in triangles. For any angle \( \theta \) in a right triangle, the sine is the ratio of the length of the opposite side to the hypotenuse.
This can be formulaically written as:\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
In this context:

• Opposite side: \( 84 \)
• Hypotenuse: \( 85 \)

Thus, the initial sine ratio is:\[ \sin \theta = \frac{84}{85} \]However, as the problem specifies \( \theta \) resides in quadrant III, you must account for the negative sign of sine in this quadrant. Therefore, the final answer for \( \sin \theta \) is:
\[ \sin \theta = -\frac{84}{85} \]The sine function is a critical tool in various fields such as sound wave analysis, signal processing, and electrical engineering. Understanding how sine values change with angles is crucial for grasping these advanced applications.