04Ctrigonometry

toc = Basic Trigonometry Definitions =

{“tri” =3, “gon” = angle, “metry” = measurement, so “trigonometry” = measurement of triangles}

Recall that Pythagoras' Theorem connects the three sides of right angled triangles.

{Initially we will only work with right angled triangles, but trigonometry can be extended to any triangle}
 * Trigonometry ** connects the angles with the sides of right angled triangles.


 * Trigonometry ** is a fundamental part of many branches of science, engineering and technology.

History
Ancient Greek mathematicians such as Euclid(Alexandria 300 BC) and Archimedes(Syracuse 250BC) studied the geometric properties of right-angled triangles and proved theorems equivalent to modern trigonometry. The modern sine function was first defined in the Indian book on Astronomy: Surya Siddhanta (age disputed) and this work was expanded on by Aryabhata (India, 500AD). Tenth century Islamic mathematicianswere using essentially modern trigonometry to solve a variety of problems. At about the same time, Chinese mathematicians developed trigonometry independently.

Hypotenuse
{pronounced: ** hi-POT-en-use ** } We already know that in any right angled triangle, the ** hypotenuse ** is
 * the side opposite the right angle
 * the longest side

Labelling Sides and Angles
We will label one of the angles as ** q ** (the Greek letter, __**theta**__).

Then:
 * the side __across the triangle__ from ** q **is called the ** Opposite Side **
 * the opposite side will be abbreviated to **OPP**
 * the shorter side __next to__ ** q ** is called the ** Adjacent Side ** {"adjacent" means "next to"}
 * the adjacent side will be abbreviated to **ADJ**
 * the side opposite the right angle is still called the ** Hypotenuse **
 * the hypotenuse will be abbreviated to **HYP**

The** sine of q **is the fraction (or ratio) formed by the length of the opposite side divided by the length of the hypotenuse.
 * Sine of Theta **

math . \qquad \text{sine of } \theta = \dfrac{\text{OPP}} {\text{HYP}} math

We use ** sin( q ) **to represent the ** sine of q ** {but we still pronounce it as "sine" (sounds like "sign")}

math . \qquad \sin \big( 30^\circ \big) = \dfrac{5}{10} = 0.5 math
 * For example ** (in the diagram on the right):

{sin( q ) is a number. It has no units.}

In any triangle where ** q **is the same (no matter how big or small the triangle is), the sine of ** q **will give the same result.

Cosine and Tangent
In the same way:

The ** cosine of q **is abbreviated to ** cos( q ) ** {pronounced "koz"} and is equal to: math . \qquad \cos \big( \theta \big) = \dfrac{\text{ADJ}} {\text{HYP}} math

The** tangent of q **is abbreviated to ** tan( q ) ** and is equal to: math . \qquad \tan \big( \theta \big) = \dfrac{\text{OPP}} {\text{ADJ}} math

SOH-CAH-TOA
So we have the 3 rules that are central to trigonometry:



The rules can be remembered using the acronym : ** SOH-CAH-TOA ** {where the letters in the acronym are the first letter of each part of the 3 rules}

Note: ** sin( q ) **, ** cos( q ) **and ** tan( q ) **are **__functions__**. They take an input ** ( q ) **and output a result.

Calculator Use
First check that your calculator is set to ** DEGREES ** mode:
 * Most calculators will show either ** DEG ** or ** D ** in their screen.
 * Many calculators have a ** DRG ** button that will cycle through the angle modes
 * Some calculators have a ** MODE ** button where you have to select ** DEG **

All scientific calculators have buttons for **SIN**, ** COS ** and ** TAN **.

To find ** sin(30) **
 * press ** [SIN] ** 30 ** [=] **

{On some calculators, you have to type **30**, __then__ press **[SIN]**}

Inverse Functions
math \text{Given that } y = \sin \big( \theta \big) \text{ is a function} math

math \text{The } \underline{\text{reverse}} \text{ function is: } \theta = \sin^{-1} \big( y \big) math

{in the same way that square root is the reverse of square}
 * sin –1 (y) ** is called the ** inverse sine **. It undoes the sine function.

** For example **, we know that: math . \qquad \sin \big( 30^\circ \big) = 0.5 math

so: math . \qquad 30^\circ = \sin^{-1} \big(0.5 \big) math

The inverse functions for cos and tan: work the same way.
 * Inverse cos : ** cos –1 (y) **
 * Inverse tan : ** tan –1 (y) **

Scientific Calculators can do Inverse Trig functions:
 * You will find them in small print above the SIN, COS and TAN buttons
 * So to use Inverse Sine, you will have to press the 2nd Function then SIN

to find ** sin –1 (0.5) **
 * For example: **
 * press ** [2nd] [SIN] ** 0.5 ** [=] **

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