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Trigonometric Identities

Tim Brzezinski, Sep 13, 2015

Pythagorean Identities
1. Pythagorean Trigonometric Identity (1)
2. Trig Identity via Areas: Dynamic Illustrator
3. Trig ID Movie (II)
4. Trig Identity via Areas (2): Dynamic Illustrator
5. Trig ID Movie (III)
Even/Odd Identities
1. Sine and Cosecant Functions (Special Property)
2. Sine Identity
3. Cosine and Secant Functions (Special Property)
4. Cosine Identity
5. Tangent and Cotangent Functions (Special Property)
6. Tangent Identity
Cofunction Identities
1. Special Trigonometric Identities: Quick Exploration
Identities from Similar Right Triangles
1. New Trig IDs From Similar Right Triangles
2. New Trig ID's from Similar Right Triangles (V2)
Sum & Difference Identities
1. Sine & Cosine of a Sum: Discovery
2. Trig ID: Geometric Investigation
3. Tangent of a Sum: Discovery
4. Tangent of a Difference: Trig Identity Puzzle + Proof!
5. Cotangent of a Sum: Discovery
6. Trig ID (2) - Geometric Investigation
7. Trig Identity (3) - Geometric Investigation
8. Double Angle Trig Identity: Proof via Equal Areas
9. Trig Identity Shown Graphically
Other Identities
1. Bizzare Trig Identity?
2. 3 Squares Problem
3. A Very Unusual Trig Identity
4. Another Popular Trig Identity
5. Trig Identity (4) - Geometric Investigation
Laws of Sines and Cosines
1. Triangle Investigation
2. Law of Sines (& Area)
3. Law of Cosines: Discovery
4. What "CO" in COsine Means
Older (Earlier) Applets
1. sin(a + b) & cos(a + b) - Discovery
2. sin(a + b) & cos(a + b) - Discovery - (Solution)
3. sin(75) & cos(75) activity

Information

Pythagorean Identities

1. Pythagorean Trigonometric Identity (1)
2. Trig Identity via Areas: Dynamic Illustrator
3. Trig ID Movie (II)
4. Trig Identity via Areas (2): Dynamic Illustrator
5. Trig ID Movie (III)

Pythagorean Trigonometric Identity (1)

This applet shows the derivation of one of the most frequently used trigonometric identities. How, specifically, does it relate to the Pythagorean Theorem?

Even/Odd Identities

1. Sine and Cosecant Functions (Special Property)
2. Sine Identity
3. Cosine and Secant Functions (Special Property)
4. Cosine Identity
5. Tangent and Cotangent Functions (Special Property)
6. Tangent Identity

Sine and Cosecant Functions (Special Property)

Suppose

is an angle drawn in standard position. Let P(x, y) be any point in the coordinate plane and let r = the distance from P to the origin. Recall

and

. Interact with the applet below for a minute or two. Then answer the questions that follow. (Be sure to move point P to various locations!)

1.

Regardless of where P lies, what is the relationship between the values of the ratios

and

?

2.

Regardless of where P lies, what is the relationship between the values of the ratios

and

?

3.

What do these 2 observations imply about the sine and cosecant functions? (Click here and/or here for a hint!)

Cofunction Identities

1. Special Trigonometric Identities: Quick Exploration

Special Trigonometric Identities: Quick Exploration

1.

Write "?" in terms of

. What do you get?

2.

In terms of the symbols "%", "$", and/or "!", what is

?

3.

In terms of the symbols "%", "$", and/or "!", what is

4.

Take your results from (1) and (3) to write a new trigonometric identity. What do you get?

5.

In terms of the symbols "%", "$", and/or "!", what is

?

6.

In terms of the symbols "%", "$", and/or "!", what is

7.

Take your results from (1) and (6) to write a new trigonometric identity. What do you get?

Identities from Similar Right Triangles

1. New Trig IDs From Similar Right Triangles
2. New Trig ID's from Similar Right Triangles (V2)

New Trig IDs From Similar Right Triangles

Recall the definitions of the 6 trigonometric functions defined at an angle drawn in standard position within the coordinate plane. (These ratios were defined in terms of x, y, & r). Interact with this diagram for a minute or two. (The 2 LARGE POINTS are moveable). Then, answer the question prompts that follow.

1.

Explain why each segment IS what it is. (Some are much easier than others). For example, how do we know the purple segment has a length = to the tangent of

?

2.

How many pairs of similar triangles do you see here? How do we know these triangles you reference are all similar to each other?

3.

You have previously learned that similar triangles have corresponding sides that are in proportion. That is, ratios of corresponding sides of similar triangles are all equal in value. Given this fact, try to author other trig identities from this picture. You can type them in the space below. Or, even better, feel free to type or use the digital pen to write them in the app below this space.

Write some new trig identities here!

Write some new trig identities here! Use this space if you run out of room above.

4.2.

5.

Sum & Difference Identities

1. Sine & Cosine of a Sum: Discovery
2. Trig ID: Geometric Investigation
3. Tangent of a Sum: Discovery
4. Tangent of a Difference: Trig Identity Puzzle + Proof!
5. Cotangent of a Sum: Discovery
6. Trig ID (2) - Geometric Investigation
7. Trig Identity (3) - Geometric Investigation
8. Double Angle Trig Identity: Proof via Equal Areas
9. Trig Identity Shown Graphically

5.1.

Sine & Cosine of a Sum: Discovery

There are 8 expressions shown off to the right. Your job is to carefully drag each expression (label) next to the segment (in the figure) whose length is given by this expression. After doing so, please answer the 2 questions that follow.

Given what you see in your diagram, write an equivalent expression for

.

Given what you see in your diagram, write an equivalent expression for

.

5.2.

5.3.

5.4.

5.5.

5.6.

5.7.

5.8.

5.9.

6.

Other Identities

1. Bizzare Trig Identity?
2. 3 Squares Problem
3. A Very Unusual Trig Identity
4. Another Popular Trig Identity
5. Trig Identity (4) - Geometric Investigation

6.1.

Bizzare Trig Identity?

Interact with this applet for a minute or two. Then answer the questions that follow.

Directions: 1) Write the measure of the blue angle as an inverse trigonometric function evaluated at a certain ratio. 2) Write the measure of the green angle as an inverse trigonometric function evaluated at a certain ratio. 3) Write the measure of the pink angle as an inverse trigonometric function evaluated at a certain ratio. 4) Write an identity that expresses the relationship among your responses to the first 3 questions.

6.2.

6.3.

6.4.

6.5.

7.

Laws of Sines and Cosines

1. Triangle Investigation
2. Law of Sines (& Area)
3. Law of Cosines: Discovery
4. What "CO" in COsine Means

7.1.

Triangle Investigation

Students: Please use this applet in conjunction with the activity you received at the beginning of today's class.

Older (Earlier) Applets

1. sin(a + b) & cos(a + b) - Discovery
2. sin(a + b) & cos(a + b) - Discovery - (Solution)
3. sin(75) & cos(75) activity

8.1.

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. After doing so, go to the following applet: http://tube.geogebra.org/m/1651617

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Trigonometric Identities

Table of Contents

Information

Pythagorean Identities

Pythagorean Trigonometric Identity (1)

Even/Odd Identities

Sine and Cosecant Functions (Special Property)

1.

2.

3.

Cofunction Identities

Special Trigonometric Identities: Quick Exploration

1.

2.

3.

4.

5.

6.

7.

Identities from Similar Right Triangles

New Trig IDs From Similar Right Triangles

1.

2.

3.

Write some new trig identities here!

Write some new trig identities here! Use this space if you run out of room above.

Sum & Difference Identities

Sine & Cosine of a Sum: Discovery

Other Identities

Bizzare Trig Identity?

Laws of Sines and Cosines

Triangle Investigation

Older (Earlier) Applets

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