Relationships Between Trigonometric Functions

Introduction

The relationships between trigonometric functions and their symmetry properties are fundamental to understanding their behavior. These relationships emerge from the unit circle, symmetry principles, and periodic properties of trigonometric graphs.

Symmetry Properties of Trigonometric Functions

Symmetry of Sine Function

The sine function exhibits symmetry about the line \( \theta = \frac{\pi}{2} \). This property can be expressed as:

\(\sin(\pi – \theta) = \sin \theta\)

Explanation:

• When \(\theta\) is measured on the unit circle, the angle \(\pi – \theta\) lies in the second quadrant.
• In the second quadrant, sine remains positive, and the vertical distance from the x-axis remains the same as for angle \(\theta\).

Symmetry of Cosine Function

The cosine function is symmetric about the y-axis. The symmetry property is given by:

\(\cos(\pi – \theta) = -\cos \theta\)

Explanation:

• In the second quadrant, the cosine of an angle is negative because the horizontal distance to the x-axis becomes negative.
• Thus, the cosine of \(\pi – \theta\) flips its sign but retains its magnitude.

Symmetry of Tangent Function

The tangent function combines the behavior of sine and cosine, showing a reflection symmetry. The property is expressed as:

\(\tan(\pi – \theta) = -\tan \theta\)

Explanation:

• Tangent is defined as \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
• In the second quadrant, sine remains positive, but cosine is negative, making tangent negative.

Unit Circle Representation

The symmetry properties of trigonometric functions can be visualized on the unit circle:

• \(\sin(\pi – \theta)\) corresponds to the y-coordinate of a point in the second quadrant, which equals \(\sin \theta\).
• \(\cos(\pi – \theta)\) corresponds to the x-coordinate in the second quadrant, which equals \(-\cos \theta\).
• \(\tan(\pi – \theta)\) combines these changes, making the tangent negative.

Relationships Between Trigonometric Functions

Reciprocal Relationships

Trigonometric functions are connected through reciprocal identities:

• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cot \theta = \frac{1}{\tan \theta}\)

These relationships allow for easy conversion between primary and reciprocal trigonometric functions.

Pythagorean Identities

The Pythagorean identities relate sine, cosine, and tangent:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(1 + \tan^2 \theta = \sec^2 \theta\)
• \(1 + \cot^2 \theta = \csc^2 \theta\)

These identities are derived using the unit circle and help solve complex trigonometric equations.

Odd and Even Functions

Trigonometric functions exhibit odd or even symmetry:

• Odd Functions: \(\sin(-\theta) = -\sin \theta\) and \(\tan(-\theta) = -\tan \theta\).
• Even Functions: \(\cos(-\theta) = \cos \theta\).

These properties simplify calculations involving negative angles.

Example Problems

Example 1: Simplify \(\sin(\pi – x)\)

Using the symmetry property of sine:

\(\sin(\pi – x) = \sin x\)

Thus, the sine of an angle in the second quadrant equals the sine of the angle in the first quadrant.

Example 2: Simplify \(\cos(\pi – x)\)

Using the symmetry property of cosine:

\(\cos(\pi – x) = -\cos x\)

The cosine becomes negative in the second quadrant.

Example 3: Verify \(\tan(\pi – x) = -\tan x\)

Start with the definition of tangent:

\(\tan(\pi – x) = \frac{\sin(\pi – x)}{\cos(\pi – x)}\)

Substitute the symmetry properties:

\(\sin(\pi – x) = \sin x\) and \(\cos(\pi – x) = -\cos x\).

Thus:

\(\tan(\pi – x) = \frac{\sin x}{-\cos x} = -\tan x\).

Links to Other Topics

• Unit Circle and Angle Measurement (SL3.5).
• Compound Angle Identities (AHL3.10).
• Odd and Even Functions (AHL2.14).