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A short class with Sarah and her students. The other ways are under More.
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Today, we will explore the graph of y = tan(θ). Can anyone tell me what they already know about tangent?
I know that tangent is related to the opposite and adjacent sides of a right triangle.
That's correct! The tangent function is indeed defined as tan(θ) = opposite/adjacent. How do you think this relates to its graph?
I guess it means the graph could have values depending on the angle?
Exactly! The values of tan(θ) will change dramatically as θ approaches certain angles, specifically where the function becomes undefined, like 90° and 270°. This leads us to the next concept: vertical asymptotes!
What are vertical asymptotes?
Great question! Vertical asymptotes are lines where the function goes to infinity, causing the graph to be undefined. For the tangent function, this occurs at every odd multiple of 90°.
So, are there specific key points on the graph we should remember?
Yes! Some critical points include: tan(0°) = 0, tan(45°) = 1, and at 90°, the function is undefined. Remembering these values can help sketch the graph.
In summary, the graph of y = tan(θ) has a repeating pattern, vertical asymptotes at odd multiples of 90°, and key points at tan(0°) and tan(45°).
Now that we've covered the basics, let's explore the properties of the tangent graph. What do you think happens to the values when we input angles greater than 90°?
I think it would keep increasing or maybe decrease?
Correct! The tangent function is periodic with a period of 180°. This means it will repeat its values every 180°. Can anyone calculate tan(180°) for me?
That would be 0, right?
Exactly! Now, since the tangent function goes to positive infinity just before reaching 90°, can you visualize how that looks on the graph?
Yes, I can see it going up and approaching a straight line.
Fantastic! That's the vertical asymptote around θ = 90°. This pattern repeats for every 180°, creating more asymptotes. Always remember, across each period, the graph will start again from tan(0°) = 0.
To summarize, the tangent graph has vertical asymptotes at odd multiples of 90°, a period of 180°, and key points we need to memorize.
Now, let’s practice sketching the graph of y = tan(θ). What is our starting point?
I think we start at the origin.
Correct! That’s where tan(0°) = 0. Now, where do we place our first asymptote?
At 90°!
Exactly! From the origin, if we sketch the curve, it will rise indefinitely as we approach 90°. What comes next after 90°?
Then it jumps back from -∞ to 0 at 180°.
Perfect! Now, as you sketch between these points, remember that you’ll have another asymptote at 270°, and it will repeat thereafter. Make sure to label these points accurately.
In summary, sketching the tangent function involves plotting key points and recognizing asymptotes at odd multiples of 90°. Excellent work today, everyone!
Defined as the ratio of the opposite side to the adjacent side in a right triangle, it varies continuously over its domain.
Tangent function exhibits vertical asymptotes at odd multiples of 90°, indicating where the function becomes undefined.
The tangent graph has a periodicity of 180°, repeating its values across intervals.
At zero, the tan is right in the middle, at 90 it’s undefined, oh what a riddle!
Imagine climbing a hill that keeps steepening until you reach 90°, where it suddenly drops away. That's how tan behaves — slopes steepen until the steepest point is too high to manage!
Use the phrase: 'The Tangent Travels Too Tall' to remember that it touches 0 at the start but climbs high towards 90°.
TAP - Tangent's Asymptote Positions: Remember the key asymptote positions at 90° and 270°.
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What is tan(0°)?
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What is the period of y = tan(θ)?
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Where are the vertical asymptotes for tan(θ)?
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A trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle, represented as tan(θ).
A line that the graph approaches but never touches or crosses, occurring where the function is undefined.
The characteristic of a function to repeat its values in regular intervals, specific to the tangent function being every 180°.
Specific values of θ where the function yields notable outputs, like tan(0°) = 0 and tan(45°) = 1.
This section focuses on the characteristics and behavior of the graph of the tangent function, y = tan(θ).
The graph of y = tan(θ) features a repeating curve with vertical asymptotes and a periodicity of 180° (or π). It is undefined at specific angles, highlighting its unique properties compared to sine and cosine functions.
The graph of the tangent function, represented as y = tan(θ), exhibits distinctive properties including a repeating curve and vertical asymptotes. Unlike the sine and cosine graphs, which oscillate between fixed amplitude values, the tangent function continues indefinitely, having a period of 180° (or π radians). The function is specifically undefined at angles such as θ = 90°, 270°, etc., where it approaches vertical asymptotes. Understanding these characteristics is crucial as they demonstrate how the tangent function behaves differently in comparison to sine and cosine functions. Key points include: