Trigonometric Functions: Sine, Cosine, and Tangent Explained
Explore the main features of the sine, cosine, and tangent functions in trigonometry. Understand how to graph each function, identify maximum and minimum values, and differentiate between their characteristics. Dive into the world of trigonometric functions and enhance your mathematical knowledge.
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10 July 2025 Trigonometric functions LO: To identify the main features of the sine function, cosine function and tangent function. www.mathssupport.org
Graphing sine function We already know the exact sine values for many angles Angle measure 90o 30o 60o 135o 150o 1 2 180o 0o 45o 120o 3 2 1 2 3 2 2 1 0 0 Sine value 2 2 2 Angle measure 225o 270o 300o 315o 330o 1 360o 390o 1 2 210o 240o 1 3 3 2 2 -1 0 Sine value If we let y = sin x we can plot these values on a graph 2 2 2 2 2 2 y 1 What if we continue finding points? The graph will repeat itself 0.5 0 270o 90o 360o x 180o 450o 540o -0.5 -1 www.mathssupport.org www.mathssupport.org
Graphing cosine function Similarly, we can do the same with the cosine values Angle measure 90o 30o 60o 1 2 135o 150o 180o 0o 45o 120o 1 3 2 3 2 -1 1 0 Cosine value 2 2 2 2 2 Angle measure 225o 270o 300o 1 2 315o 330o 360o 390o 210o 240o 1 3 2 3 2 3 1 0 Cosine value If we let y = cos x we can plot these values on a graph 2 2 2 2 2 2 y 1 What if we continue finding points? The graph will repeat itself 0.5 0 270o 90o 360o x 180o 450o 540o -0.5 -1 www.mathssupport.org www.mathssupport.org
Compare the sine and cosine functions maximum The curves are the same size and shape, only their horizontal position on the axes differ. The functions are periodic, which means that they repeat the same cycle of values over and over. The period or length of one cycle is 360o. This means that if you look at two points whose x-coordinates are 360o apart, the y-coordinate of those two points would be the same. y 1 y = sin x 0.5 amplitude x 0 270o 90o 360o 180o -0.5 -1 period maximum minimum y 1 y = cos x Both functions have a maximum value of 1. Both functions have a minimum value of -1. 0.5 amplitude x 0 270o 90o 360o 180o Each of these functions has an amplitude of 1. -0.5 The amplitude is one half the vertical distance from a maximum to a minimum -1 minimum www.mathssupport.org www.mathssupport.org
Graphing the tangent function List the tangent values in the tables Angle measure 90o 30o 60o 135o 150o 180o 0o 45o 120o 3 3 -1 1 Und. 0 0 tangent value - 3 3 3 3 Angle measure 225o 270o 300o 315o 330o 360o 390o 210o 240o 3 3 3 1 und 0 tangent value If we let y = tan x we can plot these values on a graph 3 1 3 3 3 3 What if we continue finding points? Unlike the sine and cosine functions, the tangent function does not have an amplitude. It has no maximum or minimum values. y 8 6 4 2 0 -2 270o 90o 360o x 180o 450o 540o -4 -6 -8 www.mathssupport.org www.mathssupport.org
Tangent function Like the sine and cosine functions, the tangent function is periodic. There are vertical asymptotes at x= 90o, x = 270o, x = 450o, The same cycle of values repeats between each pair of vertical asymptotes. The period of the tangent function is 180o Domain: x such that x 90o, x 270o, x 450o, Range: y 8 y = 6 Principal axis: y = 0, x-axis 4 2 0 -2 270o 90o 360o x 180o 450o 540o -4 -6 -8 www.mathssupport.org
The sine function For the sine function y 1 0.5 x 0 270o 90o 360o 630o 180o 720o 450o 540o 810o 900o -0.5 -1 Domain: Range: Period: Amplitude: Principal axis: x 1 y 1 360 1 y = 0, x-axis www.mathssupport.org www.mathssupport.org
The cosine function For the cosine function y 1 0.5 x 0 270o 90o 360o 630o 180o 720o 450o 540o 810o 900o -0.5 -1 You may also have noticed that the cosine function has the same shape as the sine function, but is translated 90 to the left cos x = sin (x + 90 ) Domain: Range: Period: Amplitude: Principal axis: x 1 y 1 360 1 y = 0, x axis www.mathssupport.org www.mathssupport.org
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