Trigonometric Functions and Identities

Welcome to the foundational concepts of trigonometry! In this section, we will explore trigonometric functions and their fundamental identities. Understanding these is crucial for solving a wide range of problems in mathematics, physics, and engineering. We'll cover the definitions, unit circle, and the core identities that form the backbone of trigonometry.

Understanding Angles and Their Measurement

Before diving into functions, let's clarify how we measure angles. Angles are typically measured in degrees or radians. A full circle is 360 degrees or 2π radians.

• Degrees: A common unit where a full circle is divided into 360 equal parts.
• Radians: A more natural unit in calculus and higher mathematics. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

The conversion between degrees and radians is essential:

To convert degrees to radians: Multiply by \( \frac{\pi}{180^\circ} \).

To convert radians to degrees: Multiply by \( \frac{180^\circ}{\pi} \).

The Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin (0,0) of a Cartesian coordinate system. It's a powerful tool for defining and visualizing trigonometric functions. For any angle \( \theta \) in standard position (vertex at the origin, initial side along the positive x-axis), the terminal side intersects the unit circle at a point \( (x, y) \).

The coordinates \( (x, y) \) of this point directly correspond to the cosine and sine of the angle:

• \( x = \cos(\theta) \)
• \( y = \sin(\theta) \)

This definition extends the concepts of sine and cosine beyond right-angled triangles to all angles, including those greater than 90 degrees or negative angles.

Defining the Six Trigonometric Functions

Using the unit circle point \( (x, y) \) for an angle \( \theta \), we define the six trigonometric functions:

• Sine (sin): \( \sin(\theta) = y \)
• Cosine (cos): \( \cos(\theta) = x \)
• Tangent (tan): \( \tan(\theta) = \frac{y}{x} \), provided \( x \neq 0 \). This is also \( \frac{\sin(\theta)}{\cos(\theta)} \).
• Cosecant (csc): \( \csc(\theta) = \frac{1}{y} \), provided \( y \neq 0 \). This is the reciprocal of sine.
• Secant (sec): \( \sec(\theta) = \frac{1}{x} \), provided \( x \neq 0 \). This is the reciprocal of cosine.
• Cotangent (cot): \( \cot(\theta) = \frac{x}{y} \), provided \( y \neq 0 \). This is the reciprocal of tangent, and also \( \frac{\cos(\theta)}{\sin(\theta)} \).

The signs of these functions depend on the quadrant in which the terminal side of the angle \( \theta \) lies.

Trigonometric Identities

Identities are equations that are true for all values of the variables for which the expressions are defined. They are fundamental tools for simplifying trigonometric expressions and solving equations.

1. Pythagorean Identities

These are derived from the Pythagorean theorem applied to the unit circle. Since for any point \( (x, y) \) on the unit circle, \( x^2 + y^2 = 1 \), and we know \( x = \cos(\theta) \) and \( y = \sin(\theta) \), we get the primary identity:

Primary Pythagorean Identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \)

From this primary identity, we can derive two others by dividing by \( \cos^2(\theta) \) and \( \sin^2(\theta) \) respectively (assuming the denominators are not zero).

Dividing by \( \cos^2(\theta) \): \( \frac{\sin^2(\theta)}{\cos^2(\theta)} + \frac{\cos^2(\theta)}{\cos^2(\theta)} = \frac{1}{\cos^2(\theta)} \) \( \implies \tan^2(\theta) + 1 = \sec^2(\theta) \)

Dividing by \( \sin^2(\theta) \): \( \frac{\sin^2(\theta)}{\sin^2(\theta)} + \frac{\cos^2(\theta)}{\sin^2(\theta)} = \frac{1}{\sin^2(\theta)} \) \( \implies 1 + \cot^2(\theta) = \csc^2(\theta) \)

2. Reciprocal Identities

These were introduced when defining the functions. They state the relationship between a function and its reciprocal.

3. Quotient Identities

These relate tangent and cotangent to sine and cosine.

Periodicity of Trigonometric Functions

A function \( f(x) \) is periodic if there exists a positive number \( P \) such that \( f(x+P) = f(x) \) for all \( x \) in the domain of \( f \). The smallest such positive number \( P \) is called the fundamental period.

The trigonometric functions repeat their values at regular intervals. This is evident from the unit circle: adding \( 2\pi \) radians (or 360°) to an angle brings you back to the same point on the circle.

• The fundamental period of \( \sin(\theta) \) and \( \cos(\theta) \) is \( 2\pi \).
• The fundamental period of \( \sec(\theta) \) and \( \csc(\theta) \) is also \( 2\pi \).
• The fundamental period of \( \tan(\theta) \) and \( \cot(\theta) \) is \( \pi \). This is because \( \tan(\theta + \pi) = \tan(\theta) \).

This periodicity can be expressed as:

• \( \sin(\theta + 2n\pi) = \sin(\theta) \)
• \( \cos(\theta + 2n\pi) = \cos(\theta) \)
• \( \tan(\theta + n\pi) = \tan(\theta) \)
• \( \csc(\theta + 2n\pi) = \csc(\theta) \)
• \( \sec(\theta + 2n\pi) = \sec(\theta) \)
• \( \cot(\theta + n\pi) = \cot(\theta) \)

where \( n \) is any integer.

Even and Odd Functions

Understanding whether a function is even or odd helps in simplifying expressions and graphing.

• A function \( f(x) \) is even if \( f(-x) = f(x) \) for all \( x \). Its graph is symmetric about the y-axis.
• A function \( f(x) \) is odd if \( f(-x) = -f(x) \) for all \( x \). Its graph is symmetric about the origin.

Let's examine the trigonometric functions:

• \( \cos(-\theta) = \cos(\theta) \), so cosine is an even function.
• \( \sec(-\theta) = \frac{1}{\cos(-\theta)} = \frac{1}{\cos(\theta)} = \sec(\theta) \), so secant is also even.
• \( \sin(-\theta) = -\sin(\theta) \), so sine is an odd function.
• \( \csc(-\theta) = \frac{1}{\sin(-\theta)} = \frac{1}{-\sin(\theta)} = -\csc(\theta) \), so cosecant is odd.
• \( \tan(-\theta) = \frac{\sin(-\theta)}{\cos(-\theta)} = \frac{-\sin(\theta)}{\cos(\theta)} = -\tan(\theta) \), so tangent is odd.
• \( \cot(-\theta) = \frac{\cos(-\theta)}{\sin(-\theta)} = \frac{\cos(\theta)}{-\sin(\theta)} = -\cot(\theta) \), so cotangent is odd.

Graphs of Trigonometric Functions

Understanding the graphs helps visualize their behavior, periodicity, and amplitude (though amplitude is more prominent in sinusoidal functions like \( A \sin(Bx+C) + D \)).

Graph of \( y = \sin(x) \):

• Starts at (0,0).
• Goes up to a maximum of 1 at \( x = \frac{\pi}{2} \).
• Crosses the x-axis at \( x = \pi \).
• Goes down to a minimum of -1 at \( x = \frac{3\pi}{2} \).
• Returns to 0 at \( x = 2\pi \).
• Periodic with period \( 2\pi \).
• Odd function.

Graph of \( y = \cos(x) \):

• Starts at its maximum value of 1 at \( x = 0 \).
• Crosses the x-axis at \( x = \frac{\pi}{2} \).
• Reaches its minimum value of -1 at \( x = \pi \).
• Crosses the x-axis again at \( x = \frac{3\pi}{2} \).
• Returns to its maximum value of 1 at \( x = 2\pi \).
• Periodic with period \( 2\pi \).
• Even function.

Graph of \( y = \tan(x) \):

• Starts at (0,0).
• Increases towards positive infinity as \( x \) approaches \( \frac{\pi}{2} \) from the left.
• Has vertical asymptotes at \( x = \frac{\pi}{2} + n\pi \) (where \( n \) is an integer).
• Crosses the x-axis at \( x = n\pi \).
• Periodic with period \( \pi \).
• Odd function.

Graphs of \( \csc(x) \), \( \sec(x) \), and \( \cot(x) \) can be sketched by considering the reciprocal of the sine, cosine, and tangent graphs, respectively. They will have vertical asymptotes where the original function is zero.

Applications of Identities

Trigonometric identities are not just theoretical; they are vital for simplifying complex expressions and solving equations.

Example 1: Simplify \( \frac{\sin(2\theta)}{2\sin(\theta)} \) This requires a double angle identity, which we will cover in the next section. For now, let's assume \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \). \( \frac{\sin(2\theta)}{2\sin(\theta)} = \frac{2\sin(\theta)\cos(\theta)}{2\sin(\theta)} = \cos(\theta) \) (provided \( \sin(\theta) \neq 0 \)).

Example 2: Prove \( \sec^2(\theta) - \tan^2(\theta) = 1 \) We know the Pythagorean identity \( 1 + \tan^2(\theta) = \sec^2(\theta) \). Rearranging this gives \( \sec^2(\theta) - \tan^2(\theta) = 1 \). This is a direct application of a known identity.

Example 3: Find \( \sin(\theta) \) if \( \cos(\theta) = \frac{3}{5} \) and \( \theta \) is in Quadrant IV. Using the primary Pythagorean identity: \( \sin^2(\theta) + \cos^2(\theta) = 1 \) \( \sin^2(\theta) + \left(\frac{3}{5}\right)^2 = 1 \) \( \sin^2(\theta) + \frac{9}{25} = 1 \) \( \sin^2(\theta) = 1 - \frac{9}{25} = \frac{16}{25} \) \( \sin(\theta) = \pm\sqrt{\frac{16}{25}} = \pm\frac{4}{5} \) Since \( \theta \) is in Quadrant IV, sine is negative. Therefore, \( \sin(\theta) = -\frac{4}{5} \).

Key Takeaways for Exams

Memorize the definitions of the six trigonometric functions in terms of \( x, y, r \) and on the unit circle \( (x, y) \).

Master the three Pythagorean identities. They are the most fundamental.

Know the reciprocal and quotient identities. They are simple but frequently used.

Understand the periodicity and the even/odd nature of the functions. This is crucial for solving equations and simplifying expressions involving angles outside the \( [0, 2\pi) \) range or negative angles.

Practice converting between degrees and radians.

The ASTC mnemonic is invaluable for determining the signs of trigonometric functions in different quadrants.