- 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:
Corresponding Parts of Similar Triangles are Proportional.
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 tangent of the angle t to be any b / a;
tan(t) = b / a.
"Similarly" we define the sine and cosine functions;
sin(t) = b / c and cos(t) = a / c.
There are also the reciprocal functions; cot(t) = 1 / tan(t) = a / b
sec(t) = 1 / cos(t) = c / a and csc(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 c from himself to the top of the building, or (2) the angle t from the horizontal up to the top of the building. Let's look at each case:
(1) You might know about the Pythagorean 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 elevation t , then we can use the appropriate trig function to find the height. Since we know the horiz dist a = 100 and we want the height b, 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 height of the "stick" is the sine of the angle; the green length (on the circle or the x-axis) is the angle in 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 amplitude A 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 period P = 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 r and the angle of elevation t (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 t and y = 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.
Circle r = 3
Cardioid r = 2 + 2 cos t
Limacon r = 1 + 3 cos t
Spiral r = t / 5
Petal Curve (odd) r = 4 cos(3 t)
Petal Curve (even) r = 3 cos(2 t)
Ellipse r = 5 / (2 + cos t)
Bach's Bug r = 2 cos 3 t + 2 sin 4 t
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