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9. Worked Examples
Interactive Audio Lesson
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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.
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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.
Overview
Short Summary
This section provides worked examples that illustrate the application of trigonometric identities and graph transformations in mathematics.
Medium Summary
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 Summary
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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Create a free accountExample 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!
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Create a free accountExample 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!
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Proving Trigonometric Identities: Understanding how to verify identities using specific angle values.
Graph Transformations: Knowing how to determine amplitude and period, and sketch transformation graphs.
Examples
Memory Aids
Interactive tools to help you remember key concepts
Rhymes
Stories
Flash Cards
Glossary
Trigonometric Identity
An equation involving trigonometric functions that holds true for all angles defined within the equation.
Pythagorean Identity
A fundamental identity stating that sin²θ + cos²θ = 1 for any angle θ.
Amplitude
The maximum height of a wave from the midline in trigonometric graphs.
Period
The length of one complete cycle in a periodic function, calculated with 360° / B where B is the coefficient of x.