7.3
Graphs of Trigonometric Functions
Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the v…
Trigonometric graphs are related to the unit circle. As a point moves counterclockwise around the circle, its vertical projection traces the sine wave.
Similarly, the horizontal projection of the point’s motion around the circle produces the cosine wave.
Sine and cosine graphs have a period of 2π, a domain of all real numbers, and a range from -1 to 1.
On the unit circle, the intersection of a vertical tangent line and a secant on the circle that passes through the origin traces the tangent function graph. The tangent function is defined everywhere except where the cosine is zero. Its range is all real numbers.
The cosecant graph repeats every 2π. It is undefined wherever the sine is zero. Its values are always less than -1 or greater than 1.
The secant graph also has a 2π period. It is undefined where cosine is zero and has the same range as cosecant.
The cotangent graph is undefined where the sine is zero and has a range of all real numbers.
Trigonometric graphs model periodic motion—like the height of a passenger on a Ferris wheel rising and falling over time traces a sine wave as the wheel rotates around its center.
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Q1: How do sine and cosine graphs relate to the unit circle?
As a point moves counterclockwise around the unit circle, its vertical projection traces the sine wave, while its horizontal projection produces the cosine wave. Both functions have a period of 2π, a domain of all real numbers, and a range from -1 to 1, creating smooth, repeating waveforms that model periodic motion.
Q2: What are the key differences between tangent and cotangent graphs?
The tangent function is defined everywhere except where cosine is zero, producing an unbounded curve with vertical asymptotes at those points and a range of all real numbers. The cotangent function is undefined where sine is zero and also has a range of all real numbers, but its asymptotes occur at different locations along the x-axis.
Q3: Why are secant and cosecant graphs undefined at certain points?
Secant and cosecant are reciprocal functions of cosine and sine, respectively. The secant graph is undefined where cosine is zero, while the cosecant graph is undefined where sine is zero. Both functions have values always less than -1 or greater than 1, and they arch away from the x-axis without crossing it.
Q4: How do trigonometric graphs model real-world periodic motion?
Trigonometric graphs represent repeating patterns found in nature and engineering. For example, the height of a passenger on a Ferris wheel rising and falling over time traces a sine wave as the wheel rotates around its center, demonstrating how trigonometric functions of real numbers describe cyclical phenomena.
Q5: What symmetries do trigonometric functions exhibit?
Sine, tangent, and cosecant are odd functions symmetric about the origin, meaning f(-x) = -f(x). Cosine and secant are even functions symmetric about the y-axis, where f(-x) = f(x). These symmetries reflect the geometric properties of the unit circle and affect how the graphs appear on both sides of the y-axis.
Q6: What determines the domain restrictions of trigonometric functions?
Domain restrictions arise from division by zero in function definitions. Tangent and secant are undefined where cosine equals zero, while cotangent and cosecant are undefined where sine equals zero. These restrictions create vertical asymptotes in the graphs and limit where each function can be evaluated.
Q7: How do the ranges of trigonometric functions differ?
Sine and cosine have bounded ranges from -1 to 1, creating waves that oscillate between these values. Tangent and cotangent have unbounded ranges of all real numbers, extending infinitely in both directions. Secant and cosecant also have unbounded ranges but are restricted to values less than -1 or greater than 1.