Goals
1. Describe and sketch the graphs of trigonometric functions.
2. Extract key information from the graphs of trigonometric functions, such as their period and roots.
3. Develop the ability to interpret graphs in real-life contexts.
Contextualization
Trigonometric functions like sine and cosine are essential when it comes to understanding the periodic patterns we see in our everyday lives, whether it's the rhythm of ocean waves or the swinging of a pendulum. Gaining insight into these graphs helps us predict behaviours and make informed forecasts, skills that are vital across many professional domains. For instance, engineers utilise these concepts to evaluate vibrations within buildings and bridges, thereby ensuring safety and stability. Similarly, in the financial sector, analysts employ these functions to map out economic cycles and forecast market trends.
Subject Relevance
To Remember!
Sine Function
The sine function is a core trigonometric function that illustrates the relationship between an angle in a right triangle and the ratio of the opposite side to the hypotenuse. Its graph forms a sine wave that fluctuates between -1 and 1, with a period of 2π.
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The sine function is periodic, repeating every 2π.
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It ranges from -1 to 1.
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The graph starts at 0, peaks at 1 (at π/2), returns to 0 (at π), drops to -1 (at 3π/2), and finishes back at 0 (at 2π).
Cosine Function
The cosine function is another fundamental trigonometric function describing the relationship between an angle in a right triangle and the ratio of the adjacent side to the hypotenuse. Its graph is akin to a sine wave but is shifted by π/2 compared to the sine function, also oscillating between -1 and 1 with a period of 2π.
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The cosine function is periodic, repeating every 2π.
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It also oscillates between -1 and 1.
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The graph begins at 1, reaches 0 (at π/2), dips to -1 (at π), returns to 0 (at 3π/2), and goes back to 1 (at 2π).
Tangent Function
The tangent function connects an angle in a right triangle to the ratio of the opposite side to the adjacent side. Unlike sine and cosine, the tangent has asymptotic behaviour, with periodicity of π, allowing values from -∞ to ∞.
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The tangent function is periodic with a period of π.
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It has vertical asymptotes where it is undefined, specifically at points (π/2 + kπ), where k is any integer.
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The tangent graph goes through 0 at integer multiples of π and increases indefinitely between the asymptotes.
Practical Applications
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Civil Engineering: Evaluating vibrations in buildings and bridges for safety.
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Economics: Modelling economic fluctuations and predicting market trends.
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Physics: Describing sound and light waves, as well as examining periodic movements like swings of a pendulum.
Key Terms
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Sine Function: Relates an angle to the ratio of the opposite side versus hypotenuse in a right triangle.
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Cosine Function: Relates an angle to the ratio of the adjacent side to the hypotenuse in a right triangle.
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Tangent Function: Relates an angle to the ratio of the opposite side to the adjacent side in a right triangle.
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Period: The interval at which a trigonometric function repeats its values.
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Amplitude: The highest or lowest value of the trigonometric function relative to the x-axis.
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Roots: Points where the trigonometric function crosses the x-axis (where the function equals zero).
Questions for Reflections
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How can understanding trigonometric graphing be of use in your future profession?
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In which other areas, besides those listed, do you think knowledge of trigonometric functions could be beneficial?
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What hurdles did you encounter while sketching the graphs of trigonometric functions, and how did you tackle them?
Hands-on Modelling of Trigonometric Graphs
Create a physical model of the sine and cosine function graphs using simple materials to help visualise period and amplitude.
Instructions
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Gather materials: graph paper, a ruler, a pencil, string, and tape.
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Draw x and y axes on the graph paper.
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Sketch a complete cycle of the sine and cosine functions, marking important points such as peaks, troughs, and roots.
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Use the string to create a tangible representation of the graphs, securing the string along the points outlined on paper.
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Present your model to your peers, explaining how you identified the periods and amplitudes of the functions.
