- All triangles have three sides. (File this fact under the "thank you Captain Obvious" category.)
- All triangles contain 180 degrees.

A^{2 }+ B^{2 }= C^{2}

**Video!**

This video walks through how to apply Pythagoras' theorem on a right triangle.

[embed] https://youtu.be/8rjvhG2pW3c [/embed]

- sin θ = opposite/hypotenuse
- cos θ = adjacent/hypotenuse
- tan θ = opposite/adjacent

- SOH - Sine is opposite/ hypotenuse
- CAH - Cosine is adjacent/hypotenuse
- TOA - Tangent is opposite/ adjacent

- sin is short for sine
- cos is short for cosine
- tan is short for tangent

**Video!**

This video walks through how to determine the angle of a right triangle when you have two sides.

[embed] https://youtu.be/qJX9l_8WThs [/embed]

**Video!**

This video walks through how to determine the other sides of a triangle when you have an angle and one side.

[embed] https://youtu.be/Aildd3S2eVI [/embed]

- I
^{2}X = Vars - E
^{2}(inductor voltage)^{ }/X = Vars - I x E (inductor voltage) = Vars

When building an impedance or power triangle, the resistive component always goes on the bottom of the triangle and the reactive component always goes on the side.

- Quadrant 1 has 0 to 90 degrees.
- Quadrant 2 has 90 to 180 degrees.
- Quadrant 3 has 180 to 270 degrees.
- Quadrant 4 has 270 to 360 degrees.

**Video!**

This video goes into greater detail about the specifics of all four quadrants.

[embed] https://youtu.be/ljX1WqEZEBw [/embed]

- cos 55°×50 = 28.7 for the X axis
- sin 55°×50 = 41 for the Y axis

- cos 40°×60 = 46 for the X axis
- sin 40°×60 = 39 for the Y axis

**Video!**

This video walks through how to convert from polar form to rectangular form.

[embed]https://youtu.be/9QX36MdH9ik[/embed]

**Video!**

This video walks through how to convert from rectangular form to polar form.

[embed] https://youtu.be/UCsovkGv24A [/embed]

- Vectors contain magnitude (resultant) and direction (angle).
- Each vector can be broken into X and Y coordinates.
- We must use a quadrant system to chart the X and Y coordinates.
- Pay attention to the polarity (what quadrant is it in?).
- Vectors can be expressed in the polar form (resultant and angle) or rectangular form (X and Y coordinates).
- Base your angle off of the X axis.
- When converting from rectangular to polar, it is
*extremely*important to pay attention to what quadrant you are in.- Quadrant 1 is the angle you calculate.
- Quadrant 2 is 180 minus the angle you calculate.
- Quadrant 3 is 180 plus the angle you calculate.
- Quadrant 4 is either 360 minus the angle you calculate, or, put a negative in front of the angle you calculate.

**Video!**

This video walks through how to add vectors heading in completely different directions.

[embed]https://youtu.be/lPnuAnbmWvg[/embed]

- The size of the magnetic field. The more flux lines there are, the more flux lines there are for the conductor to cut. The strength of flux is directly proportional to the induced voltage.
- The active length of the conductor. Active length meaning the part of the conductor that actually passes through the field. The active length is directly proportional to the induced voltage.
- The speed at which the conductor passes through the field. The faster the conductor passed through the field, the greater the voltage induced. The speed is directly proportional to the induced voltage.

e = peak voltage induced in the inductor (volts)

B = field strength between the poles (Tesla)

l = active length of conductor (meters)

v = velocity of the conductor through the field (m/sec)

Here is an example. A conductor that has an active length of 4 meters passes through a field of 5 Tesla at a speed of 15 meters per second. Determine the peak voltage induced on this conductor.(4 m)(5 T)(15 m/sec) = 300 volts peak

That's crazy! Who discovered that? The discovery of electromagnetic induction is attributed to Michael Faraday who discovered that when he passed a magnetic field through a conductor a current would flow. As long as there was motion between the field and the conductor, a voltage could be induced. This could mean the conductor passes through a field or a field passed through a conductor. Next up:- combustion engine
- hydro dam
- hand crank
- windmill

**Video!**

This video explains how a sine wave is generated in an alternator.

[embed] https://youtu.be/L_m1mAY6WZ4 [/embed]

**Video!**

This video shows all that is happening in the sine wave.

[embed] https://youtu.be/98FgWHu2eI4 [/embed]

- Average (1 alternation) = .639 × peak
- Instantaneous = sin θ × peak
- Effective (RMS) = .707 × peak

f = frequency in hertz

P = number of poles

N = rotational speed in RPM

We divide the number of poles by two because there will always be a set of two poles. You can't have a north pole without a south. We divide the RPM by 60 because we are concerned with rotations per second, not rotations per minute. The formula in Figure 56 can be combined to look like this: [caption id="attachment_289" align="aligncenter" width="97"]**Video!**

This video will walk you through how frequency is related to the RPM and the number of poles of an alternator.

[embed]https://youtu.be/7kfcpXhO19Q[/embed]

How RPM and number of poles affect frequency. video by The Electric Academy is under a Creative Commons Attribution Licence.

]]>**Pythagorean theorem/Pythagoras’ theorem:** A relation in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In simple terms, this theorem says that you can figure out any side of a right triangle as long as you have the other two sides, using the equation:
A^{2 }+ B^{2 }= C^{2
}

**resistance:** The component that opposes current in a DC circuit.

**right angle: ** An angle that equals 90 degrees.

- Pythagoras worksheet #1
- Pythagoras worksheet #2
- Trigonometry worksheet #1
- Trigonometry worksheet #2
- Trigonometry worksheet #3
- Power and Impedance triangles worksheet

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Version | Date | Change | Details |
---|---|---|---|

1.0 | December 14, 2018 | Book published in Pressbooks. | |

1.1 | January 28, 2019 | Updates made in order to publish the book in the B.C. Open Textbook Collection | Added a list of links for the print copy, created a versioning history page, and added attribution information for videos. |

- Flinn, C. (2018).
*Trigonometry and Single Phase AC Generation for Electricians*. Vancouver, B.C.: BCIT. Retrieved from https://pressbooks.bccampus.ca/trigforelectricians

Learning Objectives

- Define different types of angles
- Identify the attributes of right triangles
- Apply Pythagoras' theorem
- Demonstrate trigonometry using right triangles
- Explain impedance triangles
- Explain power triangles

See the Trigonometry section of the Appendix for worksheets that can be used in this section.

]]>Learning Objectives

- Explain vectors
- Describe a quadrant system
- Discuss the difference between polar form and rectangular form
- Demonstrate how to add vectors

Learning Objectives

- Explain electromagnetic induction
- Describe the difference between a simple alternator and practical alternator
- Discuss how a waveform is generated
- Analyze a waveform and determine how effective, instantaneous, and average values are calculated

See the AC Generation section of the Appendix for worksheets that can be used in this section.

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