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Interactive Audio Lesson
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Proving the Pythagorean Identity
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Today, we're going to prove the identity sin²θ + cos²θ = 1 using a specific angle. Can anyone tell me what sin(30°) is?
Isn't it 1/2?
Yes, exactly! So, if we square it, what do we get?
1/4!
Perfect! Now, what about cos(30°)?
It's √3/2.
Correct! Squaring that gives us what?
3/4!
Now, if we add those together, what do we find?
1/4 + 3/4 equals 1, so the identity holds!
Great job! This identity is foundational for many proofs in trigonometry. Remember: all angles θ satisfy this identity. You can think of it as a triangle’s 'Pythagorean theorem' on the unit circle.
Understanding Graph Transformations
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Now, let's explore the transformation of graphs, specifically with the function y = 3 cos(2x). What does the '3' tell us?
It indicates the amplitude, right? So it stretches the graph to a height of ±3.
Exactly! Now, can someone tell me how we determine the period of this function?
We use the formula Period = 360° / B, where B is 2.
Correct! What is our period then?
That would be 180°!
Right! So now, sketching the graph, we know one full wave occurs between 0° and 180°. Can anyone summarize what we learned about amplitude and period?
The amplitude is 3, and the graph repeats every 180°!
Well done! Remembering these transformations is key for graphing any trigonometric function.
Introduction & Overview
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Quick Overview
Standard
The worked examples guide students on how to prove fundamental trigonometric identities and demonstrate graph transformations. These practical illustrations help solidify understanding of core concepts, including proving identities and graph sketching.
Detailed
Worked Examples
In this section, we delve into practical applications of the trigonometric identities and transformations. Understanding these worked examples is essential for mastering the chapter, as they offer insight into how theoretical concepts can be applied to solve problems.
Example 1: Prove the Identity sin²θ + cos²θ = 1
To show that the Pythagorean identity holds, we use a specific angle. Let's choose θ = 30°:
- We calculate sin(30°) = 1/2, therefore
- sin²(30°) = (1/2)² = 1/4.
- Next, we calculate cos(30°) = √3/2, thus
- cos²(30°) = (√3/2)² = 3/4.
Combining these results:
- Total = 1/4 + 3/4 = 1, confirming that the identity holds true.
Example 2: Graph Transformation
Consider the function y = 3 cos(2x). We analyze its components:
- Amplitude is given as 3, which indicates the height of the wave from midline to peak.
- To find the period, we note that it is calculated through the formula Period = 360° / B, where B = 2:
- Period = 360° / 2 = 180°.
Finally, we graph this function from 0° to 180°, illustrating that one complete wave spans this interval and reaches heights of ±3.
These examples not only enhance understanding but also illustrate the continuity of trigonometric principles used to solve a variety of mathematical problems.
Audio Book
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Example 1: Proving the Identity
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Example 1: Prove the identity
sin²θ + cos²θ = 1
Let θ = 30°
• sin(30°) = 1/2 → sin²(30°) = 1/4
• cos(30°) = √3/2 → cos²(30°) = 3/4
• Total = 1/4 + 3/4 = 1 ✔
Detailed Explanation
In this example, we are asked to prove the fundamental identity sin²θ + cos²θ = 1. First, we set θ equal to 30 degrees. We find the sine and cosine of 30 degrees: sin(30°) is 1/2, and cos(30°) is √3/2. Next, we square both values: (1/2)² equals 1/4, and (√3/2)² equals 3/4. We then add these two results together: 1/4 + 3/4 gives us 1, thus confirming that sin²(30°) + cos²(30°) = 1.
Examples & Analogies
Think of this like a recipe. If you need to prove that a dish (the identity) is well-made, you gather your ingredients (values of sine and cosine), mix them properly (squaring and adding), and check your final outcome. Just like in cooking, if you follow the steps correctly, you will get the expected result!
Example 2: Graph Transformation
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Example 2: Graph Transformation
Sketch y = 3 cos(2x)
• Amplitude = 3
• Period = 360° / 2 = 180°
• The graph completes a full wave from 0° to 180° with height ±3
Detailed Explanation
In this example, we are transforming a cosine graph described by the equation y = 3 cos(2x). The amplitude is the height of the waves from the center line, so here it is 3, meaning the graph will reach up to 3 and down to -3. The coefficient before x, which is 2, affects the period of the function. The period is calculated by taking 360° and dividing it by this coefficient, resulting in a period of 180°. This tells us that the graph oscillates (completes one full cycle) from 0° to 180°.
Examples & Analogies
You can think of this transformation like adjusting the height and speed of a swing. If the swings represent the waves of the cosine graph, making it higher corresponds to increasing the amplitude (3), while making it swing faster (the period of 180°) means it goes from one extreme to the other quicker. Imagine if you adjusted a swing to move higher and swing back more quickly—that's essentially what we are visualizing with this graph transformation!
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Proving Trigonometric Identities: Understanding how to verify identities using specific angle values.
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Graph Transformations: Knowing how to determine amplitude and period, and sketch transformation graphs.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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Proving sin²θ + cos²θ = 1 by substituting θ = 30° and calculating values.
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Graphing y = 3 cos(2x), identifying amplitude and period, and sketching the graph.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When sine and cosine square, together they do care, equals one is always true, prove it, just for you!
📖 Fascinating Stories
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Imagine a triangle running a race between sine and cosine; as they square up their distances, they always sum to one!
🧠 Other Memory Gems
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Sine and Cosine keep it neat – square them up, and they can't be beat!
🎯 Super Acronyms
S² + C² = 1 (for Sine and Cosine) reminds us their squares unite!
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Trigonometric Identity
Definition:
An equation involving trigonometric functions that holds true for all angles defined within the equation.
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Term: Pythagorean Identity
Definition:
A fundamental identity stating that sin²θ + cos²θ = 1 for any angle θ.
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Term: Amplitude
Definition:
The maximum height of a wave from the midline in trigonometric graphs.
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Term: Period
Definition:
The length of one complete cycle in a periodic function, calculated with 360° / B where B is the coefficient of x.