Definitions of Sine, Cosine, and Tangent

Introduction

Key Concepts

Sine Function

The sine function is one of the primary trigonometric functions and is defined for an acute angle in a right-angled triangle as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, it is expressed as:

Where:

• Opposite is the side opposite the angle θ.
• Hypotenuse is the longest side of the right-angled triangle, opposite the right angle.

The sine function is periodic with a period of $2\pi$ radians and ranges between -1 and 1. It is widely used in modeling periodic phenomena such as sound waves, light waves, and tides.

**Example:** Consider a right-angled triangle where one of the angles is $30^\circ$, the hypotenuse is 10 units long, and the opposite side is 5 units.

Cosine Function

The cosine function is another fundamental trigonometric function, defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle. It is mathematically represented as:

Where:

• Adjacent is the side adjacent to the angle θ.
• Hypotenuse remains the longest side of the triangle.

Similar to sine, the cosine function is periodic with a period of $2\pi$ radians and has values ranging from -1 to 1. It plays a crucial role in various applications, including signal processing and navigation.

**Example:** In a right-angled triangle with an angle of $60^\circ$, a hypotenuse length of 10 units, and an adjacent side of 5 units, the cosine is calculated as:

Tangent Function

The tangent function is defined as the ratio of the sine of an angle to its cosine. In the context of a right-angled triangle, it represents the ratio of the length of the side opposite the angle to the length of the adjacent side. The mathematical formula is:

The tangent function is periodic with a period of $\pi$ radians and can take any real value from $-\infty$ to $\infty$. It is extensively used in calculus, physics, and engineering to model slopes and angles of elevation or depression.

**Example:** For a $45^\circ$ angle in a right-angled triangle with opposite and adjacent sides both measuring 7 units:

Unit Circle Definitions

Beyond right-angled triangles, sine, cosine, and tangent functions are also defined using the unit circle— a circle with a radius of one centered at the origin of a coordinate plane. For an angle θ measured from the positive x-axis:

• Sine of θ: The y-coordinate of the corresponding point on the unit circle.
• Cosine of θ: The x-coordinate of the corresponding point on the unit circle.
• Tangent of θ: The ratio of the y-coordinate to the x-coordinate, i.e., $\tan(\theta) = \frac{y}{x}$.

This perspective allows for the extension of trigonometric functions to all real numbers, enabling the study of their properties and behaviors beyond acute angles.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the occurring variables where both sides of the equation are defined. They are essential tools in simplifying expressions and solving equations in trigonometry. Key identities related to sine, cosine, and tangent include:

• Reciprocal Identities:
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$
• Pythagorean Identities:
  • $\sin^2(\theta) + \cos^2(\theta) = 1$
  • $1 + \tan^2(\theta) = \sec^2(\theta)$
  • $1 + \cot^2(\theta) = \csc^2(\theta)$
• Angle Sum and Difference Identities:
  • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
  • $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

These identities are instrumental in proving other mathematical theorems and solving complex trigonometric equations.

Graphs of Trigonometric Functions

Graphing sine, cosine, and tangent functions reveals their periodic nature and amplitude variations. Understanding these graphs is vital for visualizing how these functions behave over different intervals.

• Sine and Cosine: Both have an amplitude of 1 and a period of $2\pi$ radians. Their graphs are smooth, continuous waves oscillating between -1 and 1.
• Tangent: Has a period of $\pi$ radians and vertical asymptotes where the cosine is zero (e.g., $90^\circ$, $270^\circ$, etc.). The graph exhibits a repeating pattern of increasing and decreasing segments between asymptotes.

**Graphical Example:** - The graph of $\sin(\theta)$ starts at 0, reaches a maximum of 1 at $\frac{\pi}{2}$ radians, returns to 0 at $\pi$ radians, reaches a minimum of -1 at $\frac{3\pi}{2}$ radians, and completes its cycle at $2\pi$ radians. - The graph of $\cos(\theta)$ starts at 1, decreases to 0 at $\frac{\pi}{2}$ radians, reaches -1 at $\pi$ radians, returns to 0 at $\frac{3\pi}{2}$ radians, and completes its cycle at $2\pi$ radians. - The graph of $\tan(\theta)$ has vertical asymptotes at $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ radians, with the function increasing from $-\infty$ to $\infty$ between each pair of asymptotes.

Applications of Trigonometric Ratios

Sine, cosine, and tangent functions are pivotal in various real-world applications, including:

• Engineering: Analyzing waveforms, oscillations, and alternating current (AC) circuits.
• Architecture: Designing structures and understanding forces and angles.
• Physics: Describing periodic motions, such as pendulums and springs.
• Navigation: Calculating distances and plotting courses using trigonometric principles.
• Computer Graphics: Rendering visual effects and animations.

Mastery of these trigonometric functions enables students to tackle complex problems across diverse fields, highlighting their significance in both academic and practical contexts.

Inverse Trigonometric Functions

Inverse trigonometric functions allow the determination of an angle when the value of a trigonometric function is known. They are essential for solving equations where the angle is the unknown.

• Arcsine ($\sin^{-1}$): Returns the angle whose sine is a given number.
• Arccosine ($\cos^{-1}$): Returns the angle whose cosine is a given number.
• Arctangent ($\tan^{-1}$): Returns the angle whose tangent is a given number.

**Example:** If $\sin(\theta) = 0.5$, then $\theta = \sin^{-1}(0.5) = 30^\circ$.

Inverse trigonometric functions are restricted to specific domains to ensure they are single-valued, making them reliable tools for angle determination.

Slope of a Line and Tangent Function

The tangent function is closely related to the slope of a line in coordinate geometry. The slope (m) of a line making an angle θ with the positive x-axis is given by:

This relationship is fundamental in understanding the inclination of lines, optimizing angles for maximum efficiency, and solving geometric problems involving slopes and angles of intersection.

Trigonometric Ratios in the Unit Circle

Within the unit circle framework, trigonometric ratios extend to all real numbers, providing a comprehensive understanding of angles beyond the acute range. For any angle θ, the sine and cosine values correspond to the y and x coordinates of the point where the terminal side of the angle intersects the unit circle. The tangent is derived from these coordinates, offering insights into the behavior of trigonometric functions in different quadrants.

This approach facilitates the exploration of trigonometric identities, solving trigonometric equations, and understanding periodicity and symmetry in trigonometric graphs.

Comparison Table

Summary and Key Takeaways