*dan's trigonometry lesson *

*dan's trigonometry lesson*

* ©1997-2016 - all rights reserved - dan bach @dansmath*

**Similar Triangles and the Basic Six Trig Functions**

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:

**C**orresponding **P**arts of** S**imilar **T**riangles are **P**roportional.

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 that means 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**

When I got my teaching job at DVC (back in the 20th century), 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 *(soon) *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 horizontal distance 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°) = b/a = b/100 , so **b** = 100 tan(40°) ≈ 83 ft 11 in.

## Circles, Angles, Radians

**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)(2π)(25) = 75 π / 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 2π radius lengths, so the angle 360° is called 2π radians.

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

## Solving General Triangles, Law of Sines and Cosines

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 good old Pythagorean Theorem.

## Graphs of Trig Functions

*(using Mathematica for pictures)*

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.

**sine function ** Here's a cool sine animation (I did it in 1996, *Mathematica* to *QuickTime* to *GIFBuilder*); the height of the "stick" is the sine of the angle; the green length is the angle in radians, from 0 to 2π.

**amplitude**

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'.

**frequency**

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 = 2π = one 'wavelength.'

The bigger the B, the less x has to vary to do sin(2π) , and so the period of y = sin(Bx) is 2π/B ; in sound the frequency is 'pitch' or which note it is. Here's an animation showing this springy phenomenon.

**phase shift & horizontal shift **

The horizontal position of the graph is controlled by the B and the C in the equation y = sin(Bx – C).

The B is the *frequency* (see above), and the C is the angle or *phase* and affects the wave horizontally.

If B = 1, then y = sin(x – C) is just y = sin(x) moved to the right by C. This is because if x = C then we're doing sin(0).

For general B, write y = sin[B{x – (C/B)}]. Then if x = C/B we have sin(0). So it's a *horizontal shift* to the *right* by C/B.

The equation y = sin(2x – π/2) would have period π, phase shift π/2, but horizontal shift of only π/4. So y = 0 when x = π/4.

## Polar Coordinates

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

The use of polar coordinates can create some cool-looking graphs; shapes like circles and petal curves, limaçons, and spirals have very complicated equations in rectangular coordinates (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. There is **free graphing software **online; here are a few apps/sites to try:

www.desmos.com *(very popular, can browse and save activities)*

www.geogebra.org *(has 3D graphing capabilities)*

www.graphcalc.com *(pretty old, 2003 latest update)*

These polar graphs were done (by me in 1999) in *Mathematica*: