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A short class with Sarah and her students. The other ways are under More.
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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.
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.
When sine and cosine square, together they do care, equals one is always true, prove it, just for you!
Imagine a triangle running a race between sine and cosine; as they square up their distances, they always sum to one!
Sine and Cosine keep it neat – square them up, and they can't be beat!
S² + C² = 1 (for Sine and Cosine) reminds us their squares unite!
Term
Proving sin²θ + cos²θ
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Term
Transforming y = 3 cos(2x)
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An equation involving trigonometric functions that holds true for all angles defined within the equation.
A fundamental identity stating that sin²θ + cos²θ = 1 for any angle θ.
The maximum height of a wave from the midline in trigonometric graphs.
The length of one complete cycle in a periodic function, calculated with 360° / B where B is the coefficient of x.
This section provides worked examples that illustrate the application of trigonometric identities and graph transformations in mathematics.
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.
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.
To show that the Pythagorean identity holds, we use a specific angle. Let's choose θ = 30°:
Consider the function y = 3 cos(2x). We analyze its components:
These examples not only enhance understanding but also illustrate the continuity of trigonometric principles used to solve a variety of mathematical problems.