Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are fundamental in various fields such as physics, engineering, and computer science. The primary trigonometric functions are sine, cosine, and tangent, which can be defined using the unit circle or right triangles.

Trigonometric Functions in Right Triangles

The Triangle Definition

In a right triangle with angle $\theta$, the trigonometric functions are defined as ratios of the sides:

    |\
    | \
  a |  \ c (hypotenuse)
    |   \
    |    \
    ------
       b

Where:

opposite side (opposite to angle $\theta$) = $a$
adjacent side (next to angle $\theta$) = $b$
hypotenuse (longest side, opposite the right angle) = $c$

SOH-CAH-TOA Definitions

The mnemonic SOH-CAH-TOA helps remember the basic definitions:

SOH: $\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{a}{c}$
CAH: $\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{b}{c}$
TOA: $\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b}$

Additional Trigonometric Functions

Three reciprocal functions complete the set:

Cosecant: $\csc \theta = \frac{1}{\sin \theta} = \frac{c}{a}$
Secant: $\sec \theta = \frac{1}{\cos \theta} = \frac{c}{b}$
Cotangent: $\cot \theta = \frac{1}{\tan \theta} = \frac{b}{a}$

Fundamental Identity

From the Pythagorean theorem ($a^2 + b^2 = c^2$), we derive the fundamental trigonometric identity:

\[\sin^2 \theta + \cos^2 \theta = 1\]

This identity connects the triangle definition to the unit circle definition.

Unit Circle Definition

The Unit Circle Connection

The unit circle provides a more general definition that extends trigonometric functions beyond right triangles. A unit circle is a circle with radius 1 centered at the origin of a coordinate system.

For any angle $\theta$ measured counterclockwise from the positive x-axis:

The point where the angle intersects the unit circle has coordinates $(\cos \theta, \sin \theta)$
the x-coordinate of this point is $\cos \theta$
the y-coordinate of this point is $\sin \theta$

Why This Works

This connection between triangles and the unit circle becomes clear when we consider:

For acute angles ($0^\circ < θ < 90^\circ$): The unit circle point creates a right triangle with the x-axis
For obtuse angles ($90^\circ < θ < 180^\circ$): The definitions extend naturally using signed coordinates
For angles beyond $180^\circ$: The pattern continues, creating the periodic nature of trigonometric functions

Domain and Range

Using the unit circle definition:

Domain: All real numbers (angles can be any size)
Range of sine and cosine: $[-1, 1]$ (coordinates on unit circle)
Range of tangent: All real numbers (except at vertical asymptotes)

Sine Function

The sine function, denoted as $\sin(x)$, represents the y-coordinate of a point on the unit circle corresponding to an angle $x$ measured from the positive x-axis.

Sine Definition and Properties

Definition: $\sin(x) = \frac{\text{opposite}}{\text{hypotenuse}}$ (in right triangles) or y-coordinate on unit circle
Domain: All real numbers $(-\infty, \infty)$
Range: $[-1, 1]$
Period: $2\pi$ radians ($360^\circ$)
Symmetry: Odd function, meaning $\sin(-x) = -\sin(x)$

Sine Key Characteristics

Zero points: $\sin(x) = 0$ when $x = n\pi$ where $n$ is any integer
Maximum values: $\sin(x) = 1$ when $x = \frac{\pi}{2} + 2n\pi$
Minimum values: $\sin(x) = -1$ when $x = \frac{3\pi}{2} + 2n\pi$

Cosine Function

The cosine function, denoted as $\cos(x)$, represents the x-coordinate of a point on the unit circle corresponding to an angle $x$ measured from the positive x-axis.

Cosine Definition and Properties

Definition: $\cos(x) = \frac{\text{adjacent}}{\text{hypotenuse}}$ (in right triangles) or x-coordinate on unit circle
Domain: All real numbers $(-\infty, \infty)$
Range: $[-1, 1]$
Period: $2\pi$ radians ($360^\circ$)
Symmetry: Even function, meaning $\cos(-x) = \cos(x)$

Cosine Key Characteristics

Zero points: $\cos(x) = 0$ when $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
Maximum values: $\cos(x) = 1$ when $x = 2n\pi$
Minimum values: $\cos(x) = -1$ when $x = \pi + 2n\pi$

Tangent Function

The tangent function, denoted as $\tan(x)$, is the ratio of the sine and cosine functions:

\[\tan(x) = \frac{\sin(x)}{\cos(x)}\]

Tangent Definition and Properties

Definition: $\tan(x) = \frac{\text{opposite}}{\text{adjacent}}$ (in right triangles) or $\frac{\sin(x)}{\cos(x)}$
Domain: All real numbers except $x = \frac{\pi}{2} + n\pi$ where $n$ is any integer
Range: All real numbers $(-\infty, \infty)$
Period: $\pi$ radians ($180^\circ$)
Symmetry: Odd function, meaning $\tan(-x) = -\tan(x)$

Tangent Key Characteristics

Zero points: $\tan(x) = 0$ when $x = n\pi$ where $n$ is any integer
Vertical asymptotes: At $x = \frac{\pi}{2} + n\pi$ where $\cos(x) = 0$
No maximum or minimum values: Function approaches $\pm\infty$ at asymptotes

Characteristic Values of Trigonometric Functions

The following table shows exact values of trigonometric functions at commonly used angles:

\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{Angle } (\theta) & 0^\circ & 30^\circ & 45^\circ & 60^\circ & 90^\circ & 180^\circ & 270^\circ \\ \hline \text{Radians} & 0 & \frac{\pi}{6} & \frac{\pi}{4} & \frac{\pi}{3} & \frac{\pi}{2} & \pi & \frac{3\pi}{2} \\ \hline \sin \theta & 0 & \frac{1}{2} & \frac{\sqrt{2}}{2} & \frac{\sqrt{3}}{2} & 1 & 0 & -1 \\ \hline \cos \theta & 1 & \frac{\sqrt{3}}{2} & \frac{\sqrt{2}}{2} & \frac{1}{2} & 0 & -1 & 0 \\ \hline \tan \theta & 0 & \frac{1}{\sqrt{3}} & 1 & \sqrt{3} & \text{undefined} & 0 & \text{undefined} \\ \hline \end{array}\]

Memory Aids for Special Values

30-60-90 Triangle: Sides in ratio $1 : \sqrt{3} : 2$

\[\sin 30^\circ = \frac{1}{2}\]
\[\cos 30^\circ = \frac{\sqrt{3}}{2}\]
\[\sin 60^\circ = \frac{\sqrt{3}}{2}\]
\[\cos 60^\circ = \frac{1}{2}\]

45-45-90 Triangle: Sides in ratio $1 : 1 : \sqrt{2}$

\[\sin 45^\circ = \cos 45^\circ = \frac{\sqrt{2}}{2}\]

Quadrant Signs:

Quadrant I ($0^\circ$ to $90^\circ$): All positive
Quadrant II ($90^\circ$ to $180^\circ$): Only sine positive
Quadrant III ($180^\circ$ to $270^\circ$): Only tangent positive
Quadrant IV ($270^\circ$ to $360^\circ$): Only cosine positive

Mnemonic: "All Students Take Calculus" (A-S-T-C)

Trigonometric Laws

The Sine Law and Cosine Law are fundamental relationships that hold for any triangle, not just right triangles.

Sine Law

For any triangle with sides $a$, $b$, $c$ and opposite angles $A$, $B$, $C$ (alternatively given as opposite angles $\alpha$, $\beta$, $\gamma$):

\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]

In other words, the sine law states that the quotient of two side lengths of a triangle gives the same value as the quotient of the sines of the opposite interior angles (also: opposite angles):

\[\frac{a}{b} = \frac{\sin A}{\sin B}, \quad \frac{b}{c} = \frac{\sin B}{\sin C}, \quad \frac{c}{a} = \frac{\sin C}{\sin A}\]

This law is particularly useful when you know:

Two angles and one side (AAS or ASA)
Two sides and an angle opposite one of them (SSA)

Example: Finding a Missing Side (AAS)

Given: In triangle ABC, $A = 30^\circ$, $B = 45^\circ$, and side $a = 10$ units. Find: Side $b$.

Solution:

First find angle $C$: $C = 180^\circ - A - B = 180^\circ - 30^\circ - 45^\circ = 105^\circ$
Apply the Sine Law: $\frac{a}{\sin A} = \frac{b}{\sin B}$
Substitute: $\frac{10}{\sin 30^\circ} = \frac{b}{\sin 45^\circ}$
Solve: $b = \frac{10 \times \sin 45^\circ}{\sin 30^\circ} = \frac{10 \times \frac{\sqrt{2}}{2}}{\frac{1}{2}} = 10\sqrt{2} \approx 14.14$ units

Example: Finding a Missing Angle (SSA)

Given: In triangle ABC, side $a = 8$ units, side $b = 10$ units, and angle $A = 30^\circ$. Find: Angle $B$.

Solution:

Apply the Sine Law: $\frac{a}{\sin A} = \frac{b}{\sin B}$
Rearrange: $\sin B = \frac{b \sin A}{a}$
Substitute: $\sin B = \frac{10 \times \sin 30^\circ}{8} = \frac{10 \times \frac{1}{2}}{8} = \frac{5}{8} = 0.625$
Solve: $B = \arcsin(0.625) \approx 38.68^\circ$

Note: In SSA cases, check if a second solution exists: $B' = 180^\circ - 38.68^\circ = 141.32^\circ$. Verify which solution(s) create valid triangles by checking that all angles sum to $180^\circ$.

General Note: If the lengths of the sides $a$ and b and $A$, the angle opposite the side with length a, are given, then it is not necessary that:

\[|sin(A)\frac{b}{a}| \leq 1\]

hold. But since this should always be satisfied because it is necessary that $|\sin(B)| \leq 1$, we can see that something goes wrong for certain specifications of $a$, $b$ and $A$, that is, a triangle cannot be constructed for some choices of these quantities.

Cosine Law

For any triangle with sides $a$, $b$, $c$ and angle $C$ opposite side $c$: $c^2 = a^2 + b^2 - 2ab\cos C$

This generalizes the Pythagorean theorem: If the special case of a right triangle with vertices $A$, $B$, and $C$ is present – for example, if $C = 90^\circ$, then $\cos(C) = \cos(90^\circ) = 0$, and the term $2ab \cos(C)$ will also be 0. So in this case, the cosine theorem gives $c^2 = a^2 + b^2$, which is nothing more than the statement of Pythagoras’ theorem for right triangles.

The Cosine Law is useful when you know:

Three sides (SSS)
Two sides and the included angle (SAS)

Example: Finding a Missing Side (SAS)

Given: In triangle ABC, $a = 8$ units, $b = 6$ units, and $C = 60^\circ$. Find: Side $c$.

Solution:

Apply the Cosine Law: $c^2 = a^2 + b^2 - 2ab\cos C$
Substitute: $c^2 = 8^2 + 6^2 - 2(8)(6)\cos 60^\circ$
Calculate: $c^2 = 64 + 36 - 96 \times \frac{1}{2} = 100 - 48 = 52$
Solve: $c = \sqrt{52} = 2\sqrt{13} \approx 7.21$ units

Example: Finding a Missing Angle (SSS)

Given: In triangle ABC, $a = 5$ units, $b = 7$ units, and $c = 9$ units. Find: Angle $C$.

Solution:

Rearrange the Cosine Law: $\cos C = \frac{a^2 + b^2 - c^2}{2ab}$
Substitute: $\cos C = \frac{5^2 + 7^2 - 9^2}{2(5)(7)} = \frac{25 + 49 - 81}{70} = \frac{-7}{70} = -0.1$
Solve: $C = \arccos(-0.1) \approx 95.74^\circ$

Cosine Law and Isosceles Triangles

In isosceles triangles, where two sides are equal (say $a = b$), the Cosine Law simplifies significantly.

For angle $A$ opposite side $a$, we have:

\[\begin{aligned} \cos A &= \frac{b^2 + c^2 - a^2}{2bc} \\ &= \frac{a^2 + c^2 - a^2}{2ac} \\ &= \frac{c^2}{2ac} \\ &= \frac{c}{2a} \end{aligned}\]

For angle $B$ opposite side $b$, we have:

\[\begin{aligned} \cos B &= \frac{a^2 + c^2 - b^2}{2ac} \\ &= \frac{a^2 + c^2 - a^2}{2ac} \\ &= \frac{c^2}{2ac} \\ &= \frac{c}{2a} \end{aligned}\]

So the adjacent interior angles $A$ and $B$ are equal.

For the apex angle $C$ opposite the base $c$, we have: $C = 180 - 2A$.

This relationship can be particularly useful for finding angles or sides in isosceles triangles.

Hyperbolic sine: $\sinh(x) = \frac{e^x - e^{-x}}{2}$
Hyperbolic cosine: $\cosh(x) = \frac{e^x + e^{-x}}{2}$
Hyperbolic tangent: $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

Key Properties

Fundamental hyperbolic identity: $\cosh^2(x) - \sinh^2(x) = 1$
Initial values: $\sinh(0) = 0$, $\cosh(0) = 1$
Odd function property: $\sinh(x)$ is an odd function: $\sinh(-x) = -\sinh(x)$
Even function property: $\cosh(x)$ is an even function: $\cosh(-x) = \cosh(x)$

Relationship to Trigonometric Functions

Hyperbolic functions have many similarities to trigonometric functions:

Trigonometric identity: $\sin^2(x) + \cos^2(x) = 1$
Hyperbolic identity: $\cosh^2(x) - \sinh^2(x) = 1$

The key difference is the sign in the fundamental identity, which reflects the difference between the unit circle ($x^2 + y^2 = 1$) and the unit hyperbola ($x^2 - y^2 = 1$).

Trigonometric Functions