Trigonometry
Contents
Overview
Trigonometry is a branch of mathematics studying the relationships between lengths and angles of triangles, although it does have applications beyond triangles.
Trigonometry comes from a Greek work, which literally means “triangle measure”.
Trigonometry is heavily related to a right-triangle.
Trigonometric Functions
Trigonometric functions are defined by the relationships between different pieces of the triangle.
Three Core Trigonometric Functions
Sine, or $\sin$, is a Trigonometric function defined as the ratio of the length of the opposite leg to the length of the hypotenuse. Given the diagram, think of this as:
$$ \sin{(\theta)} = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{b}{c} $$It is read as: “The Sine of theta is …”
Trig functions are indeed functions, albeit special functions. They take an input, and return an output (one output per input, hence a function!)
Trig functions have unique characteristics, though, which will be discussed more in this Chapter.
Unlike most functions, you can omit the parentheses when expressing a trigonometric function. $\sin{(\theta)} = \sin{\theta}$
You would, however, not omit parentheses with standard functions. “f x” just doesn’t work in place of $f(x)$
Cosine, or $\cos$, is a Trigonometric function defined as the ratio of the length of the adjacent leg to the length of the hypotenuse. Given the diagram, think of this as:
$$ \cos{\theta} = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{a}{c} $$It is read as: “The Cosine of theta is…”
Tangent, or $\tan$, is the final core Trigonometric function. It is defined as the ratio of the length of the opposite leg to the length of the adjacent leg. Given the diagram, think of this as:
$$ \tan{\theta} = \frac{\text{opposite}}{\text{adjacent}} = \frac{b}{a} $$It is read as: “The Tangent of theta is…”
There is a handy mnemonic to help remember these functions:
$$ SOH \space CAH \space TOA $$Which reads as:
- SOH: Sine = Opposite / Hypotenuse
- CAH: Cosine = Adjacent / Hypotenuse
- TOA: Tangent = Opposite / Adjacent
Other Trigonometric Functions
There are other trigonometric functions and these are each defined as the reciprocal (inverse) of the original corresponding function.
Cosecant, or $\csc$, is the reciprocal of the sine function.
$$ \csc{\theta} = \frac{1}{\sin{\theta}} = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{c}{b} $$Secant, or $\sec$, is the reciprocal of the cosine function.
$$ \sec{\theta} = \frac{1}{\cos{\theta}} = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{c}{a} $$Cotangent, or $\cot$, is the reciprocal of the tangent function.
$$ \cot{\theta} = \frac{1}{\tan{\theta}} = \frac{\text{adjacent}}{\text{opposite}} = \frac{a}{b} $$