Problem 77 - Precalculus: Functions and Graphs

Question

Find the exact values of the six trigonometric functions of \(\boldsymbol{\theta}\) if \(\boldsymbol{\theta}\) is in standard position and the terminal side of \(\boldsymbol{\theta}\) is in the specified quadrant and satisfles the given condition. I; on a line having slope \(\frac{4}{3}\)

Step-by-Step Solution

Answer

The six trigonometric functions of \(\theta\) are: \(\sin(\theta) = \frac{4}{5}\), \(\cos(\theta) = \frac{3}{5}\), \(\tan(\theta) = \frac{4}{3}\), \(\csc(\theta) = \frac{5}{4}\), \(\sec(\theta) = \frac{5}{3}\), \(\cot(\theta) = \frac{3}{4}\).

Step 1: Identify the Reference Triangle

Since the terminal side of \(\theta\) is in Quadrant I and has a slope of \(\frac{4}{3}\), we can consider a right triangle where the change in \(y\) (opposite side) is 4 and the change in \(x\) (adjacent side) is 3. This forms the reference triangle for \(\theta\).

Step 2: Calculate the Hypotenuse Using Pythagorean Theorem

Use the Pythagorean theorem to find the hypotenuse: \(c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\). So, the hypotenuse of the triangle is 5.

Step 3: Determine Sine and Cosine

Use the sides of the triangle to find sine and cosine. \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{5} \). Similarly, \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{3}{5} \).

Step 4: Calculate Tangent

The tangent of \(\theta\) is the ratio of the opposite side to the adjacent side: \( \tan(\theta) = \frac{4}{3} \).

Step 5: Determine Cosecant, Secant, and Cotangent

Using reciprocal identities, find the remaining trigonometric functions: \( \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{4} \), \( \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{3} \), and \( \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{3}{4} \).

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring 90 degrees. In such triangles, the side opposite this right angle is known as the hypotenuse, and it is the longest side of the triangle. In our problem, we are dealing with a right triangle that helps determine the trigonometric functions of an angle \(\theta\) in standard position.
This triangle is defined by a slope \(\frac{4}{3}\), indicating that the opposite side (change in \(y\)) is 4 units, while the adjacent side (change in \(x\)) is 3 units.

• Adjacent Side (base): \(3\)
• Opposite Side (height): \(4\)
• Hypotenuse: \(5\)

This is a classic setup in which right triangles provide a clear framework for connecting angles with linear positions, as seen by the right angle formed at the junction of horizontal and vertical lines.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle used for finding the length of sides in a right triangle. It states that the square of the hypotenuse \(c\) is equal to the sum of the squares of the other two sides \(a\) and \(b\). Mathematically, it is written as \(a^2 + b^2 = c^2\).
In our exercise, the two sides of the triangle are known:

• \(a = 3\)
• \(b = 4\)

By applying the Pythagorean theorem, we find the hypotenuse:
\[c = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5\]
This calculation confirms our understanding of the right triangle's dimensions and is crucial for defining the trigonometric ratios accurately.

Reciprocal Identities

Reciprocal identities are essential in trigonometry for calculating some less common functions from the basic trigonometric functions. They involve the reciprocals, or "flipped" versions, of sine, cosine, and tangent. Here's how you can determine these for angle \(\theta\):

• **Sine and Cosecant:** \(\sin(\theta) = \frac{4}{5}\), so \(\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{5}{4}\)
• **Cosine and Secant:** \(\cos(\theta) = \frac{3}{5}\), thus \(\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{5}{3}\)
• **Tangent and Cotangent:** \(\tan(\theta) = \frac{4}{3}\), leading to \(\cot(\theta) = \frac{1}{\tan(\theta)} = \frac{3}{4}\)

The above identities are invaluable for transforming and understanding various trigonometric expressions, making calculations more versatile and accessible.

Quadrant I

Quadrants are essential parts of the coordinate system that help categorize angles and their terminal sides. Quadrant I is particularly important because it is where both \(x\) and \(y\) values are positive, and it spans angles from 0 to 90 degrees.
In our exercise, \(\theta\) is located in Quadrant I, indicating positive values for both sine and cosine functions. This makes our calculations straightforward and refers to a common, real-world scenario in trigonometry: using positive angle measures.
This quadrant situation helps affirm the accuracy of our trigonometric ratios, as all are inherently positive in this section of the coordinate system. Understanding which quadrant an angle falls into is crucial for interpreting and predicting the sign of trigonometric functions.