Definitions of Sine, Cosine and Tangent Using Right-Angled Triangles

Introduction

Key Concepts

Understanding Right-Angled Triangles

A right-angled triangle is a triangle in which one of the angles is exactly $90^\circ$. The side opposite the right angle is known as the hypotenuse, the longest side of the triangle, while the other two sides are referred to as the adjacent and opposite sides relative to a specific acute angle.

Definition of Sine

The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, for an angle $\theta$:

$$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} $$

Example: Consider a right-angled triangle where the side opposite $\theta$ is 3 units, and the hypotenuse is 5 units. Then:

$$ \sin(\theta) = \frac{3}{5} = 0.6 $$

Definition of Cosine

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For angle $\theta$:

$$ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$

Example: Using the same triangle, if the adjacent side to $\theta$ is 4 units, then:

$$ \cos(\theta) = \frac{4}{5} = 0.8 $$

Definition of Tangent

The tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. For angle $\theta$:

$$ \tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} $$

Example: With the opposite side as 3 units and the adjacent side as 4 units:

$$ \tan(\theta) = \frac{3}{4} = 0.75 $$>

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides of a right-angled triangle. It states that the square of the hypotenuse ($c$) is equal to the sum of the squares of the other two sides ($a$ and $b$):

$$ c^2 = a^2 + b^2 $$>

Example: If $a = 3$ units and $b = 4$ units, then:

$$ c^2 = 3^2 + 4^2 = 9 + 16 = 25 \implies c = 5 \text{ units} $$>

Unit Circles and Trigonometric Ratios

While traditionally trigonometric ratios are introduced using right-angled triangles, they can also be extended to the unit circle, where the radius is 1 unit. In this context, the sine of an angle corresponds to the y-coordinate, and the cosine corresponds to the x-coordinate of a point on the circle:

$$ \sin(\theta) = y, \quad \cos(\theta) = x $$>

This perspective allows trigonometric functions to be defined for all real numbers, not just acute angles.

Solving Right-Angled Triangles

Trigonometric ratios are powerful tools for solving right-angled triangles. Given one angle and one side length, it is possible to determine the lengths of the other sides using the definitions of sine, cosine, and tangent. For instance, if an angle $\theta$ and the hypotenuse are known, the lengths of the opposite and adjacent sides can be calculated as:

$$ \text{Opposite} = \sin(\theta) \times \text{Hypotenuse} $$> $$ \text{Adjacent} = \cos(\theta) \times \text{Hypotenuse} $$>

Inverse Trigonometric Functions

Inverse trigonometric functions allow for the determination of an angle when the ratios are known. They are essential for solving triangles when certain sides and angles are given. The inverse functions are denoted as $\sin^{-1}$, $\cos^{-1}$, and $\tan^{-1}$:

$$ \theta = \sin^{-1}\left(\frac{\text{Opposite}}{\text{Hypotenuse}}\right) $$> $$ \theta = \cos^{-1}\left(\frac{\text{Adjacent}}{\text{Hypotenuse}}\right) $$> $$ \theta = \tan^{-1}\left(\frac{\text{Opposite}}{\text{Adjacent}}\right) $$>

Applications of Trigonometric Ratios

Trigonometric ratios are widely used in various fields such as engineering, physics, architecture, and navigation. They are pivotal in solving problems involving angles and distances, modeling periodic phenomena, and analyzing wave patterns.

Advanced Concepts

Derivation of Trigonometric Ratios

The trigonometric ratios can be derived using similar triangles. Consider two right-angled triangles sharing a common acute angle $\theta$. Since the triangles are similar, the ratios of corresponding sides are equal. This similarity forms the basis for defining sine, cosine, and tangent.

Let the first triangle have sides: opposite $a$, adjacent $b$, and hypotenuse $c$. The second triangle has sides: opposite $ka$, adjacent $kb$, and hypotenuse $kc$, where $k$ is a scaling factor.

Therefore, the ratios:

$$ \sin(\theta) = \frac{a}{c}, \quad \cos(\theta) = \frac{b}{c}, \quad \tan(\theta) = \frac{a}{b} $$>

remain constant irrespective of the triangle's size, highlighting the fundamental nature of these ratios.

Proof of Pythagorean Identity

The Pythagorean Identity is a cornerstone in trigonometry, stating that:

$$ \sin^2(\theta) + \cos^2(\theta) = 1 $$>

Proof: Starting with the definitions of sine and cosine:

$$ \sin(\theta) = \frac{a}{c}, \quad \cos(\theta) = \frac{b}{c} $$>

Squaring both and adding:

$$ \sin^2(\theta) + \cos^2(\theta) = \left(\frac{a}{c}\right)^2 + \left(\frac{b}{c}\right)^2 = \frac{a^2 + b^2}{c^2} $$>

From the Pythagorean Theorem, $a^2 + b^2 = c^2$, thus:

$$ \frac{a^2 + b^2}{c^2} = \frac{c^2}{c^2} = 1 $$>

Hence:

$$ \sin^2(\theta) + \cos^2(\theta) = 1 $$>

Deriving Tangent from Sine and Cosine

Tangent can be expressed in terms of sine and cosine using the following relationship:

$$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$>

Derivation: Starting with the definitions of sine and cosine:

$$ \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}, \quad \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} $$>

Dividing sine by cosine:

$$ \tan(\theta) = \frac{\frac{\text{Opposite}}{\text{Hypotenuse}}}{\frac{\text{Adjacent}}{\text{Hypotenuse}}} = \frac{\text{Opposite}}{\text{Adjacent}} $$>

Trigonometric Identities Involving Tangent

Besides the basic definition, tangent is involved in various trigonometric identities. One such identity relates tangent to sine and cosine:

$$ 1 + \tan^2(\theta) = \sec^2(\theta) $$>

Using the Pythagorean Identity, we can derive this as follows:

$$ \sin^2(\theta) + \cos^2(\theta) = 1 \implies \frac{\sin^2(\theta)}{\cos^2(\theta)} + 1 = \frac{1}{\cos^2(\theta)} $$>

Thus:

$$ \tan^2(\theta) + 1 = \sec^2(\theta) $$>

Applications in Solving Advanced Problems

Trigonometric ratios play a crucial role in solving complex problems involving angles of elevation and depression, navigation, physics motion problems, and engineering designs. For example, determining the height of a building using the angle of elevation and the distance from the observer involves the use of tangent.

Problem: An observer is standing 50 meters away from the base of a tower. If the angle of elevation to the top of the tower is $30^\circ$, find the height of the tower.

Solution: Using tangent:

$$ \tan(30^\circ) = \frac{\text{Height}}{50} $$> $$ \text{Height} = 50 \times \tan(30^\circ) = 50 \times \frac{\sqrt{3}}{3} \approx 28.87 \text{ meters} $$>

Interdisciplinary Connections

Trigonometric ratios are not confined to mathematics alone. They are pivotal in physics for analyzing wave functions, in engineering for design and stress analysis, in computer science for computer graphics and gaming, and in economics for modeling cyclical trends. For instance, in electrical engineering, sine and cosine functions model alternating current (AC) waveforms.

Inverse Trigonometric Functions and Their Applications

Inverse trigonometric functions enable the determination of angles when the trigonometric ratios are known, which is essential in various applications such as robotics, where determining angles of joints is necessary for movement, and in aerospace engineering for calculating trajectories.

Example: If $\tan(\theta) = 1$, then:

$$ \theta = \tan^{-1}(1) = 45^\circ $$>

Trigonometric Form of Complex Numbers

In complex analysis, trigonometric ratios are used to express complex numbers in polar form. A complex number $z = a + bi$ can be represented as:

$$ z = r(\cos(\theta) + i\sin(\theta)) $$>

where $r = \sqrt{a^2 + b^2}$ is the modulus and $\theta = \tan^{-1}\left(\frac{b}{a}\right)$ is the argument of the complex number.

Conclusion of Advanced Concepts

The exploration of sine, cosine, and tangent using right-angled triangles lays the groundwork for a multitude of advanced mathematical and practical applications. From proving fundamental identities to solving real-world problems, the depth and versatility of these trigonometric ratios are evident in their wide-ranging uses across various disciplines.

Comparison Table

Trigonometric Ratio	Definition	Formula	Common Applications
Sine ($\sin$)	Ratio of the opposite side to the hypotenuse	$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$	Determining heights, physics wave analysis
Cosine ($\cos$)	Ratio of the adjacent side to the hypotenuse	$\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$	Engineering design, computer graphics
Tangent ($\tan$)	Ratio of the opposite side to the adjacent side	$\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$	Navigation, architecture

Summary and Key Takeaways