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Learn it your way
A short class with Sarah and her students. The other ways are under More.
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Today, we're going to look at the graph of y = sin(θ). Who can tell me the shape of this graph?
Isn't it like a wave?
Exactly! We call it the sine curve because it has a wave-like form. Can anyone tell me what the amplitude of the sine function is?
Is it 1?
Yes, correct! The amplitude is the height of the peaks from the centerline, which is y = 0. Now, what about its period?
The period is 360° or 2π, right?
Well done! That means it completes one full wave in that interval. Now let’s go over some key points on this graph. What is sin(0°)?
It's 0!
Correct again! And what about sin(90°)?
That one is 1.
Great job! And we continue this way for 180°, 270°, and 360°. Remember these key points; they'll help when graphing.
In conclusion, the sine graph is wave-like with an amplitude of 1, a period of 360°, and key points at 0°, 90°, 180°, 270°, and 360°.
Next, let's discuss the graph of y = cos(θ). What does this graph resemble?
It looks a lot like the sine graph, right?
Exactly! It’s also a wave-like curve. It shares the same amplitude of 1, but let's talk about its key points. What can we say about cos(0°)?
I think it's 1.
Correct! And what about cos(90°)?
That would be 0.
Right! Now, continue with cos(180°) for me!
It’s -1.
Well done! And finally, cos(360°)?
That’s back to 1 again.
All very correct! Just as with the sine function, the cosine function has a period of 360°. Remember, while their shapes are similar, the peaks of the cosine graph start at 1.
To summarize, the cosine graph has an amplitude of 1, a period of 360°, with key points at 0°, 90°, 180°, 270°, and 360°.
Now let's move on to the graph of y = tan(θ). This one is quite different from sine and cosine. Who can tell me what the shape of this graph looks like?
It has steep curves that seem to go up and down, right?
Exactly! The tangent graph features curves with vertical asymptotes. Can anyone remind me at what angles are those asymptotes located?
They happen at 90° and 270°, don’t they?
Correct! The tangent function is undefined at those points. What's the period of this function?
It’s 180° or π!
Great! And let's look at some key points, like tan(0°). What is that value?
That one equals 0.
Right! What about tan(45°)?
That's 1.
Correct again! So in summary, the tangent graph has a unique shape with a period of 180°, undefined points at 90° and 270°, and key points at 0° and 45°.
Sine goes up, then down, with peaks at one, it's the wave of fun!
Imagine a surfer riding a wave, going up to a peak of 1 and back down to 0, then under to -1, repeating this every full cycle from shore to shore!
Use 'SOH-CAH-TOA' for sine, cosine, and tangent definitions!
Remember: Sine is S (start at zero), Cosine is C (start high at one), Tangent is T (touch the undefined lines!).
Term
Sine Function
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Term
Cosine Function
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Term
Tangent Function
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This section covers the graphs of sine, cosine, and tangent functions, highlighting their shapes, key points, and characteristics such as amplitude and period.
In this section, students explore the graphical representations of the trigonometric functions sine, cosine, and tangent. The graphs illustrate wave-like patterns, amplitude, periods, and specific key points of these functions. Understanding these graphical elements is crucial for applications in various fields such as engineering and physics.
In this section, we explore the graphical representations of the three primary trigonometric functions: sine, cosine, and tangent.
Understanding these graphs is essential for modeling periodic phenomena in various scientific fields and developing a solid foundation for further mathematical study.