Trigonometry

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# Trigonometry - PowerPoint PPT Presentation

Trigonometry. Instant Trig. Trigonometry is math, so many people find it scary It’s usually taught in a one-semester high-school course However, 95% of all the “trig” you’ll ever need to know can be covered in 15 minutes And that’s what we’re going to do now. 20°. 44°. 30°. 120°. 68°.

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## PowerPoint Slideshow about 'Trigonometry' - Sophia

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### Trigonometry

Instant Trig
• Trigonometry is math, so many people find it scary
• It’s usually taught in a one-semester high-school course
• However, 95% of all the “trig” you’ll ever need to know can be covered in 15 minutes
• And that’s what we’re going to do now

20°

44°

30°

120°

68°

68°

20°

44°

30°

68°

+ 130°

+ 68°

180°

180°

• The angles of a triangle always add up to 180°
We only care about right triangles

A right triangle is one in which one of the angles is 90°

Here’s a right triangle:

We call the longest side the hypotenuse

We pick one of the other angles--not the right angle

We name the other two sides relative to that angle

Here’s the angle

we are looking at

Right triangles

Here’s the

right angle

hypotenuse

opposite

If you square the length of the two shorter sides and add them, you get the square of the length of the hypotenuse

32 + 42 = 52, or 9 + 16 = 25

5 = sqrt(9 + 16)

The Pythagorean Theorem
There are few triangles with integer sides that satisfy the Pythagorean formula

3-4-5 and itsmultiples (6-8-10, etc.)are the best known

5-12-13 and its multiples form another set

25 + 144 = 169

hyp

opp

5-12-13
Since a triangle has three sides, there are six ways to divide the lengths of the sides

Each of these six ratios has a name (and an abbreviation)

Three ratios are most used:

sine = sin = opp / hyp

cosine = cos = adj / hyp

tangent = tan = opp / adj

The other three ratios are redundant with these and can be ignored

The ratios depend on the shape of the triangle (the angles) but not on the size

hypotenuse

hypotenuse

opposite

opposite

Ratios
With these functions, if you know an angle (in addition to the right angle) and the length of a side, you can compute all other angles and lengths of sides

If you know the angle marked in red (call it A) and you know the length of the adjacent side, then

tan A = opp / adj, so length of opposite side is given byopp = adj * tan A

cos A = adj / hyp, so length of hypotenuse is given byhyp = adj / cos A

hypotenuse

opposite

Using the ratios
Java methods in java.lang.Math
• public static double sin(double a)
• If a is zero, the result is zero
• public static double cos(double a)
• public static double sin(double a)
• If a is zero, the result is zero
• However: The angle amust be measured in radians
• Fortunately, Java has these additional methods:
• public static double toRadians(double degrees)
• public static double toDegrees(double radians)

hypotenuse

opposite

The hard part
• If you understood this lecture, you’re in great shape for doing all kinds of things with basic graphics
• Here’s the part I’ve always found the hardest:
• Memorizing the names of the ratios
• sin = opp / hyp
• cos = adj / hyp
• tan = opp / adj
Mnemonics from wikiquote
• The formulas for right-triangle trigonometric functions are:
• Sine = Opposite / Hypotenuse
• Cosine = Adjacent / Hypotenuse
• Tangent = Opposite / Adjacent
• Mnemonics for those formulas are:
• Some Old Horse Caught Another Horse Taking Oats Away
• Saints On High Can Always Have Tea Or Alcohol

hyp

opp

Drawing a “Turtle”

You want to move h units in theangle  direction, to (x1, y1):

You are at: (x, y)

So you make a right triangle...

And you label it...

And you compute: x1 = x + adj = x + hyp * (adj/hyp) = x + hyp * cos 

y1 = y - opp = y - hyp * (opp/hyp) = y - hyp * sin 

This is the first point in your “Turtle” triangle

Find the other points similarly...