Trigonometric graphs
Understanding Trigonometric Graphs
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Trigonometric graphs provide a visual representation of the sine (sin), cosine (cos), and tangent (tan) functions.
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Trigonometric graphs allow you to visualise the periodic nature of these functions, which repeat their values in regular intervals or cycles.
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Knowing how to read and interpret these graphs is vital in understanding future mathematical subjects, including pre-calculus and calculus.
Sine and Cosine Graphs
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Sin and cos graphs are both continuous and periodic. They display a wave-like pattern.
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Both graphs have a period of 360 degrees (or 2π radians), meaning they repeat every 360 degrees.
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The range of the sine and cosine functions is -1 to 1. Neither function ever goes above 1 or below -1.
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The sine graph starts at 0, rises to 1 at 90 degrees, drops back to 0 at 180 degrees, goes down to -1 at 270 degrees, and then back up to 0 at 360 degrees.
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The cosine graph starts at 1, drops to 0 at 90 degrees, goes down to -1 at 180 degrees, rises back to 0 at 270 degrees, and then back up to 1 at 360 degrees.
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Changing the frequency of sine or cosine graph changes how often the function repeats. It compresses or stretches the graph horizontally.
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Changing the amplitude of a sine or cosine graph changes how high and low the function goes. It stretches or compresses the graph vertically.
Tangent Graphs
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The tangent graph shows the tangent function, which is a ratio of sine to cosine.
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The tangent graph is not continuous and has vertical asymptotes, which are lines the graph approaches but never reaches.
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The tangent function has a period of 180 degrees (or π radians).
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At -90 and 90 degrees (and any multiple of 180 degrees from these), the tangent function is undefined. This is where the vertical asymptotes are.
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The range of the tan function is all real numbers. It can go as high or as low as it wants to.
Practical Tips
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It’s beneficial to familiarise yourself with the general shape and characteristics of these graphs.
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Always pay attention to the scale of the graph, especially if it has been distorted.
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Practice sketching these graphs and working with different frequencies and amplitudes.
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Mastering these concepts is key to understanding more advanced mathematical ideas in future.