Problem 17 - Calculus

Question

Draw \(\cos \theta\) and \(\sec \theta\) on the same graph. Find all points where \(\cos \theta=\sec \theta\).

Step-by-Step Solution

Answer

The points are at \( \theta = 2\pi k \) and \( \theta = (2k + 1)\pi \), where \( k \) is an integer.

Step 1: Understand the Functions

First, we need to understand what the cosine and secant functions represent. Cosine, denoted as \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Secant, \( \sec \theta \), is the reciprocal of cosine, so \( \sec \theta = \frac{1}{\cos \theta} \).

Step 2: Set Up the Equation

Now, set the equation \( \cos \theta = \sec \theta \). This implies \( \cos \theta = \frac{1}{\cos \theta} \).

Step 3: Solve the Equation

To solve \( \cos \theta = \frac{1}{\cos \theta} \), multiply both sides by \( \cos \theta \) to eliminate the fraction: \( \cos^2 \theta = 1 \).

Step 4: Find Possible Solutions

Solve \( \cos^2 \theta = 1 \). This equation gives us two possible solutions: \( \cos \theta = 1 \) and \( \cos \theta = -1 \).

Step 5: Determine \( \theta \) Values

For \( \cos \theta = 1 \), \( \theta = 2\pi k \) where \( k \) is an integer. For \( \cos \theta = -1 \), \( \theta = (2k + 1)\pi \) where \( k \) is an integer. These are the angles where the functions meet.

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos \theta \), is a fundamental trigonometric function that relates an angle \( \theta \) in a right-angled triangle to the ratios of two sides. Specifically, it is defined as the ratio of the adjacent side (the side next to the angle) over the hypotenuse (the longest side of the triangle).

• The input for the cosine function is an angle, often measured in radians or degrees.
• The output is a real number between -1 and 1.
• \( \cos \) is periodic with a period of \( 2\pi \), meaning the values repeat every \( 2\pi \) radians.

This function is particularly important because it describes the horizontal coordinate of a unit circle. In trigonometric graphs, \( \cos \theta \) is a smooth, wave-like curve that oscillates between 1 and -1. This oscillatory behavior explains why \( \cos \theta \) is used in various applications including sound waves, light waves, and electrical currents.

Secant Function

The secant function, \( \sec \theta \), is another crucial trigonometric function. It can be thought of as a sibling to the cosine function. Although not as commonly used as sine or cosine, the secant function holds its own place in trigonometry, especially in calculus and advanced geometry.

• It is defined as the reciprocal of the cosine function: \( \sec \theta = \frac{1}{\cos \theta} \).
• Since it is the reciprocal, wherever \( \cos \theta = 0 \), \( \sec \theta \) is undefined, leading to vertical asymptotes in its graph.
• \( \sec \theta \) shares the same period as \( \cos \theta \), which is \( 2\pi \).

Graphically, the secant function appears as a series of upward and downward curves that extend towards infinity near the points where the cosine function crosses the x-axis (at angles where \( \cos \theta = 0\)). This characteristic makes the secant useful in various amplitude and frequency-related calculations.

Trigonometric Identities

Trigonometric identities are useful mathematical relationships that simplify the analysis and calculations of angles and lengths. They involve the trigonometric functions (such as sine, cosine, and secant) and are instrumental when solving trigonometric equations and verifying equivalences.

• The Pythagorean identity is one of the most fundamental: \( \cos^2 \theta + \sin^2 \theta = 1 \).
• This identity is particularly useful for transforming expressions and finding unknown trigonometric values.
• The reciprocal identities like \( \sec \theta = \frac{1}{\cos \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \) simplify calculations involving complex fractions.

In the context of exercises like the one presented, identities allow us to convert equations into solvable forms, as seen in the transformation \( \cos \theta = \frac{1}{\cos \theta} \) leading to \( \cos^2 \theta = 1 \). Understanding and applying these identities is crucial in trigonometry, calculus, and physics.