Problem 4 - Precalculus

Question

The terminal side of an angle \(\theta\) in standard position passes through values of the six trigonometric functions for angle \(\theta\) $$\left(\frac{4}{7}, \frac{2}{3}\right)$$

Step-by-Step Solution

Answer

The trigonometric functions are: \( \sin \theta = \frac{42}{3\sqrt{340}} \), \( \cos \theta = \frac{84}{7\sqrt{340}} \), \( \tan \theta = \frac{7}{6} \), \( \csc \theta = \frac{3\sqrt{340}}{42} \), \( \sec \theta = \frac{7\sqrt{340}}{84} \), \( \cot \theta = \frac{6}{7} \).

Step 1: Identify Coordinates of the Point

The coordinates given are \( \left( \frac{4}{7}, \frac{2}{3} \right) \). These represent a point \((x, y)\) on the terminal side of angle \(\theta\) in standard position. Here, \(x = \frac{4}{7}\) and \(y = \frac{2}{3}\).

Step 2: Calculate the Hypotenuse

The hypotenuse \(r\) can be calculated using the Pythagorean theorem: \( r = \sqrt{x^2 + y^2} \). Substituting the values gives \( r = \sqrt{\left(\frac{4}{7}\right)^2 + \left(\frac{2}{3}\right)^2} \).

Step 3: Simplify the Hypotenuse Calculation

Calculate the squares and their sum: \( r = \sqrt{\left(\frac{16}{49}\right) + \left(\frac{4}{9}\right)} \). To add these, find a common denominator: \( \frac{16 \times 9 + 4 \times 49}{49 \times 9} = \frac{144 + 196}{441} = \frac{340}{441} \). Therefore, \( r = \sqrt{\frac{340}{441}} = \frac{\sqrt{340}}{21} \).

Step 4: Calculate Sine and Cosine

Sine is calculated as \( \sin \theta = \frac{y}{r} = \frac{\frac{2}{3}}{\frac{\sqrt{340}}{21}} = \frac{42}{3\sqrt{340}} \). Cosine is \( \cos \theta = \frac{x}{r} = \frac{\frac{4}{7}}{\frac{\sqrt{340}}{21}} = \frac{84}{7\sqrt{340}} \).

Step 5: Calculate Tangent and Reciprocal Functions

Tangent is \( \tan \theta = \frac{y}{x} = \frac{\frac{2}{3}}{\frac{4}{7}} = \frac{14}{12} = \frac{7}{6} \). For the reciprocal functions: - Cosecant \( \csc \theta = \frac{1}{\sin \theta} = \frac{3\sqrt{340}}{42} \)- Secant \( \sec \theta = \frac{1}{\cos \theta} = \frac{7 \sqrt{340}}{84} \)- Cotangent \( \cot \theta = \frac{1}{\tan \theta} = \frac{6}{7} \)

Key Concepts

Angle in Standard Position

An angle in standard position is a fundamental concept in trigonometry. It refers to an angle placed on a coordinate plane where its vertex is located at the origin point (0,0) and the initial side of the angle lies along the positive x-axis. This positioning serves as the default or standard for measuring angles.

• The terminal side of the angle is the ray that rotates from the initial position, measuring the extent of the angle.
• When dealing with such angles, it's crucial to understand that any point on the terminal side can represent the trigonometric calculations of this angle.
• In the given exercise, the point \((\frac{4}{7}, \frac{2}{3})\) lies on the terminal side of the angle \( \theta \).

Angles positioned in this way help standardize how we approach calculations with trigonometric functions, making it easier to evaluate function values across a consistent framework.

Pythagorean Theorem

The Pythagorean theorem is a mathematical principle that allows us to find the length of one side of a right-angled triangle when the lengths of the other two sides are known. The theorem is super useful in trigonometry for calculating the hypotenuse, which is the longest side of the triangle.
For a right triangle with legs \(x\) and \(y\), and hypotenuse \(r\), the theorem is represented as:
\[ r^2 = x^2 + y^2 \]

• To find \(r\), the hypotenuse in the example, you plug \(x = \frac{4}{7}\) and \(y = \frac{2}{3}\) into the equation.
• Calculate the hypotenuse exactly: \( r = \sqrt{\left(\frac{4}{7}\right)^2 + \left(\frac{2}{3}\right)^2} \).
• Completing the algebra gives \( r = \frac{\sqrt{340}}{21} \).

This theorem ensures that trigonometric ratios are accurate by providing the true length of the triangle sides relative to each angle.

Trigonometric Ratios

Trigonometric ratios are the basic functions used to relate the angles of a triangle to its side lengths. These are essential tools in trigonometry, and each ratio is derived from a specific relationship within a right triangle.

• **Sine (\(\sin \theta\))**: This is the ratio of the opposite side to the hypotenuse. In the example, \( \sin \theta = \frac{y}{r} \).
• **Cosine (\(\cos \theta\))**: This ratio compares the adjacent side to the hypotenuse. It's calculated as \( \cos \theta = \frac{x}{r} \).
• **Tangent (\(\tan \theta\))**: This ratio links the opposite side to the adjacent side, expressed as \( \tan \theta = \frac{y}{x} \).

Together, these ratios allow you to understand the spatial characteristics of triangles and work out the values for various angles precisely.

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the basic trigonometric functions. Each of them involves taking the reciprocal of one of the primary functions. They further extend the understanding of trigonometric relationships, useful for solving equations where direct trigonometric methods are not sufficient.

• **Cosecant (\(\csc \theta\))**: This is the reciprocal of sine, represented as \( \csc \theta = \frac{1}{\sin \theta} \).
• **Secant (\(\sec \theta\))**: This function is the reciprocal of cosine, written as \( \sec \theta = \frac{1}{\cos \theta} \).
• **Cotangent (\(\cot \theta\))**: This is the reciprocal of tangent, expressed as \( \cot \theta = \frac{1}{\tan \theta} \).

These reciprocal functions are particularly useful in contexts where ratios need to be inverted or re-framed to solve trigonometric problems effectively.