Graphs of Trigonometric Functions
Graphs of Basic Sine and Cosine Functions
The domains of $ f(t) = t $ and $ f(t) = t $ are identical: all real numbers, R. The ranges of these functions are also identical: the interval $ [-1, 1] $ . The graph of $ u = t $ is shown in Fig. 21-1.
The graph of $ u = t $ is shown in Fig. 21-2.
Properties of the Basic Graphs
The function $ f(t) = t $ is periodic with period $ 2$ . Its graph repeats a cycle, regarded as the portion of the graph for $ 0 t $ . The graph is often referred to as the basic sine curve. The amplitude of the basic sine curve, defined as half the difference between the maximum and minimum values of the function, is 1. The function $ f(t) = t $ is also periodic with period $ 2$ . Its graph, called the basic cosine curve, also repeats a cycle, regarded as the portion of this graph for $ 0 t $ . The graph can also be thought of as a sine curve with amplitude 1, shifted left by an amount $ /2 $ .
Graphs of Other Sine and Cosine Functions
The graphs of the following are variations of the basic sine and cosine curves.
- GRAPHS OF u = A $ t $ AND u = A $ t $ . The graph of u = A $ t $ for positive A is a basic sine curve, but stretched by a factor of A, hence with amplitude A, referred to as a standard sine curve. The graph of
u = A $ t $ for negative A is a standard sine curve with amplitude |A|, reflected with respect to the x-axis, called an upside-down sine curve. Similarly, the graph of u = A $ t $ for positive A is a basic cosine curve with amplitude |A|, referred to as a standard cosine curve. The graph of u = A $ t $ for negative A is a standard cosine curve with amplitude |A|, reflected with respect to the x-axis, called an upside-down cosine curve.
GRAPHS OF $ u = bt $ AND $ u = bt $ (b positive). The graph of $ u = bt $ is a standard sine curve, compressed by a factor of b with respect to the x-axis, hence with period $ 2/b $ . The graph of $ u = bt $ is a standard cosine curve with period $ 2/b $ .
GRAPHS OF \(u = \sin(t - c)\) AND \(u = \cos(t - c)\). The graph of \(u = \sin(t - c)\) is a standard sine curve shifted to the right lcl units if \(c\) is positive, shifted to the left lcl units if \(c\) is negative. The graph of \(u = \cos(t - c)\) is a standard cosine curve shifted to the right lcl units if \(c\) is positive, shifted to the left lcl units if \(c\) is negative. \(c\) is referred to as the phase shift. (Note: The definition of phase shift is not universally agreed upon.)
GRAPHS OF $ u = t + d $ AND $ u = t + d $ . The graph of $ u = t + d $ is a standard sine curve shifted up |d| units if d is positive, shifted down |d| units if d is negative. The graph of $ u = t + d $ is a standard cosine curve shifted up |d| units if d is positive, shifted down |d| units if d is negative.
GRAPHS OF $ u = A (bt - c) + d $ AND $ u = A (bt - c) + d $ display combinations of the above features. In general, assuming A, b, c, d positive, the graphs are standard sine and cosine curves, respectively, with amplitude A, period $ 2/b $ , phase shift c/b, shifted up d units.
EXAMPLE 21.1 Sketch a graph of u = 3 cost.
The graph (Fig. 21-3) is a standard cosine curve with amplitude 3 and period 2π.
EXAMPLE 21.2 Sketch a graph of $ u = -2 2t $ .
The graph (Fig. 21-4) is an upside-down sine curve with amplitude |-2| = 2 and period 2π/2 = π.
Graphs of the Other Trigonometric Functions
- TANGENT. The domain of the tangent function is $ {t R t /2 + 2n, 3/2 + 2n} $ and the range is R. The graph is shown in Fig. 21-5.
- SECANT. The domain of the secant function is $ {tRt/2+2n,3/2+2n} $ and the range is $ (-,-1][1,) $ . The graph is shown in Fig. 21-6.
- COTANGENT. The domain of the cotangent function is $ {t R t n} $ and the range is R. The graph is shown in Fig. 21-7.
- COSECANT. The domain of the cosecant function is $ {t R t n} $ and the range is $ (-, -1] [1, ) $ . The graph is shown in Fig. 21-8.
SOLVED PROBLEMS
21.1. Explain the properties of the graph of the sine function.
Recall that $ t $ is defined as the y-coordinate of the point $ P(t) $ obtained by proceeding a distance $ |t| $ around the unit circle from the point $ (1,0) $ . (See Fig. 21-9.) As t increases from 0 to $ /2 $ , the y-coordinate of $ P(t) $ increases from 0 to 1; as t increases from $ /2 $ through $ $ to $ 3/2 $ , y decreases from 1 through 0 to -1; as t increases from $ 3/2 $ to $ 2$ , y increases from -1 to 0. (See Fig. 21-10.) This represents one cycle or period of the sine function; since the sine function is periodic with period $ 2$ , the cycle shown in Fig. 21-10 is repeated as t increases from $ 2$ to $ 4$ , $ 4$ to $ 6$ , and so on. For negative t, the cycle is also repeated as t increases from $ -2$ to 0, from $ -4$ to $ -2$ , and so on.
21.2. Explain how to sketch a graph of $ u = A (bt - c) + d $ .
Determine amplitude and shape: Amplitude = |A| If A is positive, the curve is a standard sine curve; if A is negative, the curve is an upside-down sine curve. The maximum height of the curve is $ d + |A| $ , the minimum is $ d - |A| $ .
Determine period and phase shift: Since $ T $ goes through one cycle in the interval $ 0 T $ , $ (bt - c) $ goes through one cycle in the interval $ 0 bt - c $ ; that is, $ c/b t (c + 2)/b $ . The graph is a standard (or upside-down) sine curve with period $ 2/b $ and phase shift c/b.
Divide the interval from c/b to $ (c + 2)/b $ into four equal subintervals and sketch one cycle of the curve. For positive A, the curve increases from a height of d to its maximum height in the first subinterval, decreases to d in the second and to its minimum height in the third, then increases to d in the fourth. For negative A, the curve decreases from a height of d to its minimum height in the first subinterval, increases to d in the second and to its maximum height in the third, then decreases to d in the fourth.
Show the behavior of the curve in further cycles as desired.
21.3. Explain the properties of the graph of the cosine function
Recall that cost is defined as the x-coordinate of the point $ P(t) $ obtained by proceeding a distance $ |t| $ around the unit circle from the point $ (1,0) $ . (See Fig. 21-11.) As t increases from 0 through $ /2 $ to $ $ , the x-coordinate of $ P(t) $ decreases from 1 through 0 to -1; as t increases from $ $ through $ 3/2 $ , to $ 2$ , x increases from -1 through 0 to 1. (See Fig. 21-12). This represents one cycle or period of the cosine function; since the cosine function is periodic with period $ 2$ , the cycle shown in Fig. 21-12 is repeated as t increases from $ 2$ to $ 4$ , $ 4$ to $ 6$ , and so on. For negative t, the cycle is also repeated as t increases from $ -2$ to 0, from $ -4$ to $ -2$ , and so on.
21.4. Explain how to sketch a graph of $ u = A (bt - c) + d $ .
Determine amplitude and shape: Amplitude = |A|. If A is positive, the curve is a standard cosine curve; if A is negative, the curve is an upside down cosine curve. The maximum height of the curve is $ d + |A| $ , the minimum is $ d - |A| $ .
Determine period and phase shift: Since $ T $ goes through one cycle in the interval $ 0 T $ , $ (bt - c) $ goes through one cycle in the interval $ 0 bt - c $ , that is, $ c/b t (c + 2)/b $ . The graph is a standard (or upside-down) cosine curve with period $ 2/b $ and phase shift c/b.
Divide the interval from c/b to $ (c + 2)/b $ into four equal subintervals and sketch one cycle of the curve. For positive A, the curve decreases from its maximum height to a height of d in the first subinterval, and to its minimum height in the second, then increases to a height of d in the third subinterval and to its maximum height in the fourth. For negative A, the curve increases from its minimum height to a height of d in the first subinterval, and to its maximum height in the second, then decreases to a height of d in the third subinterval and to its minimum height in the fourth.
Show the behavior of the curve in further cycles as desired.
21.5. Sketch a graph of $ u = 6 t $
Amplitude = 6. The graph is a standard sine curve. Period $ = 2/2 = 4$ . Phase shift = 0; d = 0. Divide the interval from 0 to $ 4$ into four equal subintervals and sketch the curve with maximum height 6 and minimum height -6. See Fig. 21-13.
21.6. Sketch a graph of $ u = 3 t + 2 $
Amplitude = 3. The graph is a standard cosine curve. Period $ = 2= 2 $ . Phase shift = 0; d = 2. Divide the interval from 0 to 2 into four equal subintervals and sketch the curve with maximum height 5 and minimum height -1. See Fig. 21-14.
21.7. Sketch a graph of $ u = 2 (5t - ) $
Amplitude = 2. The graph is a standard sine curve. Period = $ 2/5 $ . Phase shift = $ /5 $ ; d = 0. Divide the interval from $ /5 $ to $ 3/5 $ (= phase shift + one period) into four equal subintervals and sketch the curve with maximum height 2 and minimum height -2. See Fig. 21-15.
21.8. Sketch a graph of $ u = -(3t + ) + $
Amplitude = \(\frac{1}{2}\). The graph is an upside down cosine curve. Period = \(\frac{2\pi}{3}\). Phase shift = \(\left(-\frac{\pi}{4}\right) \div 3 = -\frac{\pi}{12}\). Divide the interval from \(- \frac{\pi}{12}\) to \(\frac{7\pi}{12}\) (= phase shift + one period) into four equal subintervals and sketch the curve with maximum height 2 and minimum height 1. See Fig. 21-16.
21.9. Sketch a graph of $ u = | t | $
The graph is the same as the graph of $ u = t $ in the intervals for which $ t $ is positive, that is, $ (0, ) $ , $ (2, 3) $ , $ (-2, -) $ , and so on. In the intervals for which $ t $ is negative, that is, $ (, 2) $ , $ (- , 0) $ , and so on, since $ |t| = -t $ in these intervals, the graph is the same as the graph of $ u = -t $ , that is, the graph of $ u = t $ reflected with respect to the t axis (Fig. 21-17).
21.10. Explain the properties of the graph of the tangent function
Recall that $ t $ is defined as the ratio y/x of the coordinates of the point $ P(t) $ obtained by proceeding a distance l/l around the unit circle from the point $ (1,0) $ . (See Fig. 21-18.) As t increases from 0 to $ /4 $ this ratio increases from 0 to 1; as t continues to increase from $ /4 $ toward $ /2 $ , the ratio continues to increase beyond all bounds, as $ t /2^{-} $ (approaches from the left), $ t $ . Thus, the line $ t = /2 $ is a vertical asymptote for the graph. Since tangent is an odd function, the graph has origin symmetry, the line $ t = -/2 $ is also a vertical asymptote, and the curve is as shown in Fig. 21-19 for the interval $ (- /2, /2) $ . Since the tangent function has period $ $ , the graph repeats this cycle for the intervals $ (/2, 3/2) $ , $ (3/2, 5/2) $ , $ (-3/2, -/2) $ , and so on.
21.11. Sketch a graph of $ u = (t - /3) $
The graph is the same as the graph of $ u = t $ shifted $ /3 $ units to the right, and has period $ $ . Since $ T $ goes through one cycle in the interval $ -/2 < T < /2 $ , $ (t - /3) $ goes through one cycle in the interval $ -/2 < t - /3 < /2 $ , that is, $ -/6 < t < 5/6 $ . Sketch the graph in this interval and repeat the cycle with period $ $ .
21.12. Explain the properties and sketch the graph of the secant function
Since $ t $ is the reciprocal of $ t $ , it is convenient to understand the graph of the secant function in terms of the graph of the cosine function: the secant function is even, has period $ 2$ , and has vertical asymptotes at the zeros of the cosine function, that is, at $ t = /2 + 2n $ or $ 3/2 + 2n $ , n any integer. Where $ t = 1 $ , $ t = 1 $ , that is, for $ t = 0 + 2n $ , n any integer. Where $ t = -1 $ , $ t = -1 $ , that is, for $ t = + 2n $ , n any integer. As t increases from 0 to $ /2 $ , $ t $ decreases from 1 to 0; thus, $ t $ increases from 1 beyond all bounds; as t increases from $ /2 $ to $ $ , $ t $ decreases from 0 to -1, thus, $ t $ increases from unboundedly large and negative to -1. To graph u = $ t $ , sketch a graph of u = $ t $ (shown as a dotted curve in Fig. 21-21), mark vertical asymptotes through the zeros, sketch the secant curve increasing from 1 beyond all bounds at t increases from 0 to $ /2 $ and increasing from unboundedly large and negative to -1 as t increases from $ /2 $ to $ $ . Use the even property of the function to draw the portion of the graph for the interval from $ -$ to 0, then the periodicity of the function to indicate further portions of the graph.
21.13. Sketch a graph of $ u = t t $
Since $ |t| $ , $ 0 |t| |t| |t| $ , thus $ -|t| |t| |t| |t| $ , for all t. Thus the graph of u = t sin t lies between the lines u = t and u = -t. Moreover, since $ t t = 0 $ at $ t = n$ and $ t t = t $ at $ t = n+ /2 $ , the graph of u = t sin t has t intercepts at $ t = n$ and touches the lines at $ t = n+ /2 $ . The function is an even function; the graph is as shown in Fig. 21-22.
SUPPLEMENTARY PROBLEMS
21.14. State the amplitude and period of (a) $ u = t $ ; (b) $ u = 2 t - 4 $ .
Ans. (a) amplitude = 1, period = 2; (b) amplitude = 2, period = $ 2$ .
21.15. Sketch a graph of (a) $ u = t $ ; (b) $ u = 2 t - 4 $ .
Ans. (a) Fig. 21-23; (b) Fig. 21-24.
21.16. State the amplitude, period, and phase shift of (a) $ u = 2t $ ; (b) $ u = -2(t - ) + 4 $ .
Ans. (a) amplitude=\(\frac { 1 } { 3 }\) , period=π, phase shift=0; (b) amplitude=2, period=6π, phase shift=3π.
21.17. Sketch a graph of (a) $ u = 2t $ ; (b) $ u = -2 (t - ) + 4 $ .
Ans. (a) Fig. 21-25; (b) Fig. 21-26.
21.18. State the period of (a) $ u = t $ ; (b) $ u = -2t $ .
Ans. (a) $ 2$ ; (b) $ $
21.19. Sketch a graph of (a) $ u = t $ ; (b) $ u = -2t $ .
Ans. (a) Fig. 21-27; (b) Fig. 21-28.
