dansmath > lessons > precalculus > **trig**

- Basic Trig Functions
- Solving Right Triangles
- Circles and Radians
- Solving General Triangles ANIMATED !
- Graphs, Waves, and Music ANIMATED !
- Polar Coordinates
- Intermed.Algebra, Functions & Graphs
- (c) 1997-2001 Dan Bach and B & L Math Enterprises; all rights reserved. Download for personal use only.

Similar Triangles and the Basic Six Trig Functions[ top of page ] (orig. 5/98)The word "similar" means the same shape, but maybe not the same size. Like a scale model. What we need here is "proportionality." That's where CPSTP comes from:

CorrespondingParts ofSimilarTriangles areProportional.We use this similarity proportion to prove that in a right triangle, the ratio of, say, the height to the base depends only on the shape of the triangle and not the size of it.

So b / a = y / x , and we can define the

tangentof the angle t to be any b / a;

tan(t) = b / a."Similarly" we define the

sineandcosinefunctions;

sin(t) = b / candcos(t) = a / c.There are also the reciprocal functions;

cot(t) = 1 / tan(t) = a / b

sec(t) = 1 / cos(t) = c / aandcsc(t) = 1 / sin(t) = c / b.

Solving Right Triangles[ top of page ] (5/98)When I got my job at DVC, my dad asked me what I was teaching. I told him "Calculus, Statistics, and Trig." He replied, "Trigonometry? Oh, I learned that. Isn't it stuff like, you're 100 feet from a building, how tall is the building?" I said you might need to know some more information, Dad, like maybe an angle.

In the little story, my dad also needed to know either (1) the distance

cfrom himself to the top of the building, or (2) the angletfrom the horizontal up to the top of the building. Let's look at each case:

(1)You might know about thePythagorean Theorem, which relates the three sides of a right triangle:a^2 + b^2 = c^2.Click here for a cool picture proof of this theorem!

- If we know the distances a and c, then in this case we can just use algebra to find the height, b. For example, if a = 100 ft and c = 120 ft, then using b^2 = c^2 - a^2,
- b = -/(14400 - 10000) = -/4400 :=: 66 ft 4 in.

(2)If we know the angle of elevationt, then we can use the appropriate trig function to find the height. Since we know the horiz dista= 100 and we want the heightb, we use the function that relates these: the tangent function: tan(t) = height / horiz. =b/a.

- So if the angle is maybe 40 degrees, then the height is given by
- tan(40 deg) =
b/a=b/100 , so . . .b= 100 tan(40 deg) :=: 83 ft 11 in.

Circles, Angles, Radians[ top of page ] (11/98)How high off the ground are you and how far have you gone if you're in a Ferris Wheel (like a giant bicycle wheel with people in it) in which the people at the top are 50 feet off the ground, and you've gone 3/8 of the way around?

- You could draw a picture and measure, but let's figure out the angle and use trig. The angle all the way around is 360* (that means degrees for now) , so 3/8 of that is 135*, figure out the sine of 135* (it's -/2 / 2 ; think about it) and add 25 of those to the 25-ft radius, and get
- h = 25 + 25 -/2 :=: 42.7 ft off the ground.
But how far have you traveled? That's an "arclength" question; it's 3/8 of the circumference of a 25-ft radius circle; so s = (3/8)(2 Pi)(25) = 75 Pi / 4 :=: 58.9 ft.

Notice that the angle itself can be given in terms of the arclength compared to the radius; "all the way around" the circle is 2 Pi radius lengths, so the angle 360* is called 2 Pi radians.

One radian= 360* / (2 Pi)= 180* / Pi, or about 57.3 degrees.

Solving General Triangles, Law of Sines and Cosines[ top of page ] (4/99)Ever wonder why, in any triangle, that the longest side is always opposite the biggest angle? There's a way to make it more exact, called the

Law of Sines:In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :

(sin A) / a = (sin B) / b = (sin C) / c .(Show me the proof!)Also, what about the third side of a triangle if you know two sides and the included angle? If the angle were 90 degrees you could use the Pythagorean Theorem, but in general there's the

Law of Cosines:In a triangle ABC, with angles A, B, C opposite sides of lengths a, b, and c :

c^2 = a^2 + b^2 - 2ab cos C .(Show animation!)This is cool because if C is a right angle then cos C = 0 and we get the old Pythm.

Graphs of Trig Functions(using Mathematica for pictures)[ top of page ] (11/98)Linear equations like y = mx + b give lines, and quadratic equations, y = ax^2 + bx + c, give parabolas, but the trig functions sine and cosine give waves (like a side view of a slinky) that go on forever.

Here's a cool sine animation (I did it from Mathematica to QuickTime to GIF Builder); the

heightof the "stick" is thesineof the angle; the greenlength(on the circle or the x-axis) is theanglein radians, from 0 to 2Pi.

Amplitude and Frequency(1/01)

- The above animated graph showed the function y = sin(x), which is y = 1 sin(x).
- The following animation shows y = A sin(x) , for A = -3, -2, -1, 0, 1, 2, and 3.
- You can see the effect the
amplitudeA has. In sound waves this is the 'volume'.

- On the other hand, y = sin(x) is y = sin(1 x).
- The graphs of y = sin(B x) all have amplitude 1 but differ in
frequency.- Notice the graph of y = sin(x) has
periodP = 2 pi; one 'wavelength.'- The bigger the B, the less x has to vary to do sin(2 pi) , and so the period
- of y = sin(B x) is 2 pi / B ; in sound the frequency is 'pitch' or which note it is.
- Here's another animation showing this springy phenomenon.

(Graphs: Mathematica -> QuickTime -> GIF Builder)

Polar Coordinates[ top of page ] (4/99)In navigating a ship, you wouldn't hear the cry, "there's an iceberg 3 miles east and 4 miles north of us!", you'd hear "there's an iceberg 5 miles away, at a heading of about 53 degrees north of east!" The radar screen is set up to give angles and radial distances, not x and y coordinates.

We call the radial distance

rand the angle of elevationt(for the Greek letter "theta.") The pair of numbers(r,t)form the polar coordinates of the point. See the picture for the simple relation between(r,t)and(x,y):By definition, cos t = x / r and sin t = y / r , so

x = r cos tandy = r sin t .

Graphs in Polar Coordinates(1/01)

- The use of polar coords can create some cool looking graphs; shapes like circles
- and petal curves, limacons, and spirals have very complicated equations in rectang
- coords
(x,y), but much simpler, more elegant relations in polar(r,t).- Here are a few examples; I encourage you to try your own using a graphing
- calculator or computer; free graphing software is at www.graphcalc.com ;
- these were done in Mathematica.

Circler = 3

Cardioidr = 2 + 2 cos t

Limaconr = 1 + 3 cos t

Spiralr = t / 5

Petal Curve (odd)r = 4 cos(3 t)

Petal Curve (even)r = 3 cos(2 t)

Ellipser = 5 / (2 + cos t)

Bach's Bugr = 2 cos 3 t + 2 sin 4 t

**That's it for now. Check back often for new stuff!****Click below for other topics, or visit the ask dan page!**

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