Textpotential Trigonometry Video Podcast

Solving Trig Equations Part 1 - Example 6

Fri, 8 Feb 2008 21:20:45 -0600

Solving Trig Equations Part 1 - Example 6

Solving Trig Equations Part 1 - Example 5

Thu, 7 Feb 2008 21:20:38 -0600

Solving Trig Equations Part 1 - Example 5

Solving Trig Equations Part 1 - Example 4

Wed, 6 Feb 2008 21:20:29 -0600

Solving Trig Equations Part 1 - Example 4

Solving Trig Equations Part 1 - Example 3

Tue, 5 Feb 2008 21:20:18 -0600

Solving Trig Equations Part 1 - Example 3

Solving Trig Equations Part 1 - Example 2

Mon, 4 Feb 2008 21:18:51 -0600

Solving Trig Equations Part 1 - Example 2

Solving Trig Equations Part 1 - Example 1

Sun, 3 Feb 2008 21:17:17 -0600

Solving Trig Equations Part 1 - Example 1

Trig - Identities - Deriving the Sum and Difference Identities - Part 1

Sat, 2 Feb 2008 21:22:52 -0600

The Sum and Difference Identities For Sine and Cosine:
cos(alpha +/- beta) == cos(alpha)cos(beta) -/+ sin(alpha)sin(beta)
sin(alpha +/- beta) == sin(alpha)cos(beta) +/- cos(alpha)sin(beta)

Trig - Identities - Cofunction Identities

Fri, 1 Feb 2008 21:19:27 -0600

The Cofunction Identities:
cos(theta) == sin(Pi/2 - theta) tan(theta) == cot(Pi/2 - theta) sec(theta) == csc(Pi/2 - theta)
sin(theta) == cos(Pi/2 - theta) cot(theta) == tan(Pi/2 - theta) csc(theta) == sec(Pi/2 - theta)

Reciprocal, Quotient, and Pythagorean Identities Example 4 Solution 2

Fri, 25 Jan 2008 21:22:43 -0600

Reciprocal, Quotient, and Pythagorean Identities Example 4 Solution 2

Reciprocal, Quotient, and Pythagorean Identities Example 4 Solution 1

Thu, 24 Jan 2008 22:08:12 -0600

Reciprocal, Quotient, and Pythagorean Identities Example 4 Solution 1

Reciprocal, Quotient, and Pythagorean Identities Example 3

Wed, 23 Jan 2008 22:15:51 -0600

Reciprocal, Quotient, and Pythagorean Identities Example 3

Reciprocal, Quotient, and Pythagorean Identities Example 2

Tue, 22 Jan 2008 22:14:50 -0600

Reciprocal, Quotient, and Pythagorean Identities Example 2

Reciprocal, Quotient, and Pythagorean Identities Example 1

Mon, 21 Jan 2008 22:10:19 -0600

Reciprocal, Quotient, and Pythagorean Identities Example 1

Trig - Identities - Reciprocal, Quotient, and Pythagorean Identities

Sun, 20 Jan 2008 13:46:31 -0600

Reciprocal Identities

sin(x) == 1/csc(x) <=> csc(x) == 1/sin(x)
cos(x) == 1/sec(x) <=> sec(x) == 1/cos(x)
tan(x) == 1/cot(x) <=> cot(x) == 1/tan(x)

Quotient Identities

tan(x) == sin(x)/cos(x) == sec(x)/csc(x)
cot(x) = =cos(x)/sin(x) == csc(x)/sec(x)

Pythagorean Identities

sin^2(x) + cos^2(x) == 1
tan^2(x) + 1 == sec^2(x)
1 + cot^2(x) == csc^2(x)

Learn to derive these from the ratios based on right triangles and learn to use them when manipulating equations.

Example Derivation:

Pythagorean Identity: x^2 + y^2 == r^2
Divide both sides by r^2 : (x/r)^2 + (y/r)^2 == (r/r)^2 == 1
Replace ratios with their representative trig functions: sin^2(x) + cos^2(x) == 1

What Is An Identity?

Sat, 19 Jan 2008 13:43:02 -0600

An identity is an equation that holds true when any value is plugged into the variable where the expressions on each side of the equation are defined.

Example Identity 1: sin^2(x) + cos^2(x) == 1

Plug in any value for x above and the left hand side = right hand side.

Example Identity 2: tan^2(x) + 1 == sec^2(x)

Plug in any value for x above where tan(x) and sec(x) are both defined and the left hand side = right hand side.

Notation - Equality Symbols

Fri, 18 Jan 2008 18:19:10 -0600

• Conditionally Equal to
• Identically (Unconditionally) Equal to
• Defined to be Equal to

Trig - Introduction - The Unit Circle

Wed, 16 Jan 2008 17:51:17 -0600

You can use the unit circle to evaluate your trig functions. For a given angle, theta, you need to know the corresponding (x, y) coordinate on the unit circle where the terminal ray of the angle intersects the circle.

You can find many points on the unit circle using the info we learned about the 45/45 degree right triangle with hypotenuse 1 and the 30/60 degree right triangle with hypotenuse 1. Refer to the “The Two Most Important Triangles” movie for a refresher on those triangles.

The 6 trig functions are just the ratios of the 3 sides of any right triangle formed inside the angle

cos(theta) := x/r sec(theta) := r/x
sin(theta) := y/r csc(theta) := r/y
tan(theta) := y/x cot(theta) := x/y

Since these ratios are the same for any chosen radius, the choice of the unit circle (radius = 1) simplifies the ratios (plug in 1 for r everywhere) and the trig functions become

cos(theta) := x sec(theta) := 1/x
sin(theta) := y csc(theta) := 1/y
tan(theta) := y/x cot(theta) := x/y

Trig - Introduction - The Two Most Important Triangles

Tue, 15 Jan 2008 17:49:59 -0600

These two triangles give you all of basic trigonometry. You can put them in the unit circle to find a corresponding point on the circle for a given angle.

Trig - Introduction - Area of a Sector

Mon, 14 Jan 2008 18:17:32 -0600

Area of a sector = 1/2*r^2*theta

Trig - Introduction - Circular Motion - Linear and Angular Speed

Sun, 13 Jan 2008 18:14:40 -0600

Linear Speed := v = arclength/time = s/t

Example: 30 miles/hour

Angular Speed := w = angle/time = theta/t

w is the greek letter “omega” and this letter is conventionally used for angular speed. Notice angular speed has units radians/time but radians are dimension-less units so “radians per time” is really just “per time”. These are the units of frequence, sometimes referred to as “Hertz”. We will see w used again later when graphing the trig functions and the same letter is used because w will have a relationship with the frequency of the function (Refer to the “Amplitude and Frequency Modulation” movie).

Example: 3 laps/minute

Note: A lap or a revolution is an angle. 1 lap = 1 revolution = 360 degrees = 2*Pi radians.

Convert between Linear Speed and Angular Speed using v = r*w

Trig - Introduction - Angles - Converting Degrees To Radians

Sat, 12 Jan 2008 18:11:04 -0600

180 degrees = Pi radians

<=> 1 degree = Pi/180 radians

<=> 1 radian = 180/Pi degrees

Convert between degrees and radians by multiplying by a “smart one” such as (Pi radians)/(180 degrees)

30 degrees = Pi/6 radians

45 degrees = Pi/4 radians

60 degrees = Pi/3 radians

Trig - Introduction - Angles - Radians

Fri, 11 Jan 2008 18:06:40 -0600

theta := s/r radians

Radians are dimension-less units.

When you draw a circle with any radius around an angle, the angle’s measure in radians is simply the ratio of the arclength to the radius. This ratio is the same for any circle drawn around the vertex of the angle.

If a circle is drawn around an angle where the center of the circle is the vertex of the angle, we call the angle a CENTRAL ANGLE.

Pi = 3.1415... is a number

Pi happens to be the ratio of a circle’s circumference to it’s diameter: C/D = 2*Pi*r/2*r = Pi

Trig - Introduction - Angles - Degree, Minute, Second Notation

Thu, 10 Jan 2008 18:08:41 -0600

1 Degree = 60 Minutes = 60’
1 Minute = 60 Seconds = 60”

Trig - Introduction - Angles - Degrees

Wed, 9 Jan 2008 18:04:43 -0600

There are 360 degrees in an angle formed by 1 rotation.

Trig - Introduction - Angles - The Basics

Tue, 8 Jan 2008 17:55:02 -0600

We can relate to angles by drawing a circle centered around them and looking a ratios of arc length to radius or look at right triangles drawn inside the angle and looking at ratios of any two sides of those right triangles.

Notation - Factorials

Sat, 5 Jan 2008 17:47:51 -0600

Factorials are represented with an “!” symbol. They tell you to multiply the number before the ! by every positive integer before it. For instance,

5! = 5*4*3*2*1 = 120

In general, for any positive integer n, we have

n! = n*(n - 1)*(n - 2)*...*3*2*1

Notation - Sums with Sigma

Fri, 4 Jan 2008 17:44:51 -0600

Convenient notation for a bunch of things with something in common added together.

Review - Proof of the Pythagorean Theorem

Thu, 3 Jan 2008 22:30:43 -0600

A visual proof of the Pythagorean Theorem from the book “Proofs Without Words”.

Review - Quadratic to Doughnuts

Wed, 2 Jan 2008 22:28:48 -0600

Interpreting mathematical quantities. The letters are more than just numbers. You can interpret the mathematical expressions and use analogies to relate to the math.

Review - Solving Linear Equations

Tue, 1 Jan 2008 22:23:01 -0600

A linear equation has your variable to only the first power.

Example: 3x -7 = 1

A linear function: f(x) = 3x + 1

The graph of a linear function is a line. There is only one root (the x-intercept).

Notation - “Implies” and “If and Only If” Symbols

Sat, 22 Dec 2007 05:29:28 -0600

A => B
“A implies B”
“If A then B”
“A therefore B”

Example:
If A means “It is raining outside” and B means “my lawn is wet” then A => B because rain makes the lawn wet. We do not have the opposite statement though ( B does not imply A) because a wet lawn could be caused by other things than rain, such as a water sprinkler system, or it could have rained earlier in the day, or your pet could have whizzed on the lawn.

A <=B
“A is implied by B”

If it happens that A => B and B => A then both of these statements can be expressed as a single statement:
A <=> B
read “A if and only if B” or abbreviate the if and only if part with iff:
“A iff B”
Since A implies B and B implies A, this means that statements A and B are logically equivalent (A is true only when B is true and B is true only when A is true).