Trigonometry: Identities

Tim Brzezinski, Mallory Cumpstone, Aug 18, 2016

Table of Contents

  1. Pythagorean Identities
    1. Pythagorean Trigonometric Identity (1)
    2. Trig ID Movie (II)
    3. Trig ID Movie (III)
  2. Identities from Similar Right Triangles
    1. Trig Identities from Similar Right Triangles (I)
    2. Trig Identities from Similar Right Triangles (II)
  3. Sum & Difference Identities
    1. Sine & Cosine of a Sum: Discovery
    2. Tangent of a Sum: Discovery
    3. sin(75) & cos(75) activity
  4. Older (Earlier) Applets
    1. sin(a + b) & cos(a + b) - Discovery
    2. sin(a + b) & cos(a + b) - Discovery - (Solution)
  5. Other Identities
    1. A Very Unusual Trig Identity
    2. What "CO" in COsine Means
    3. Bizzare Trig Identity?
    4. Law of Cosines: Discovery

1.

Pythagorean Identities

  • 1. Pythagorean Trigonometric Identity (1)
  • 2. Trig ID Movie (II)
  • 3. Trig ID Movie (III)

1.1.

Pythagorean Trigonometric Identity (1)

This applet shows the derivation of one of the most frequently used trigonometric identities. [br][br]How, specifically, does it relate to the Pythagorean Theorem?
– Tim Brzezinski

1.2.

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1.3.

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2.

Identities from Similar Right Triangles

  • 1. Trig Identities from Similar Right Triangles (I)
  • 2. Trig Identities from Similar Right Triangles (II)

2.1.

Trig Identities from Similar Right Triangles (I)

See below:
– Tim Brzezinski

2.2.

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3.

Sum & Difference Identities

  • 1. Sine & Cosine of a Sum: Discovery
  • 2. Tangent of a Sum: Discovery
  • 3. sin(75) & cos(75) activity

3.1.

Sine & Cosine of a Sum: Discovery

[color=#000000]There are 8 expressions shown off to the right.  [/color]Your job is to carefully drag each expression (label) next to the segment (in the figure) whose length is given by this expression.  [br][br]After doing so, please answer the 2 questions that follow.
Given what you see in your diagram, write an equivalent expression for [math]\sin\left(\alpha+\beta\right)[/math].
Given what you see in your diagram, write an equivalent expression for [math]\cos\left(\alpha+\beta\right)[/math].
– Tim Brzezinski

3.2.

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3.3.

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4.

Older (Earlier) Applets

  • 1. sin(a + b) & cos(a + b) - Discovery
  • 2. sin(a + b) & cos(a + b) - Discovery - (Solution)

4.1.

sin(a + b) & cos(a + b) - Discovery

Directions are given in the applet below.
Use your results to write 2 new trigonometric identities from what you now see. [br][b][color=#c51414]After doing so, go to the following applet: http://tube.geogebra.org/m/1651617[/color][/b]
– Tim Brzezinski

4.2.

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5.

Other Identities

  • 1. A Very Unusual Trig Identity
  • 2. What "CO" in COsine Means
  • 3. Bizzare Trig Identity?
  • 4. Law of Cosines: Discovery

5.1.

A Very Unusual Trig Identity

See below: [br][br](Questions follow the applet.)
Questions:[br][br]1) Write a conditional trigonometric identity (i.e. a trigonometric identity written as an "if-then" statement) from what you've observed.[br]2) [b]*Challenge*[/b] Prove this conditional trigonometric identity.
– Tim Brzezinski

5.2.

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5.3.

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5.4.

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