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Concept 01 / 11Trigonometry

Graphs of Trigonometric Functions, Periods and the Range of a sin x + b cos x

Graphs, domains, ranges and periods of the six functions, periods of sin(ax + b) and of sums and powers, expressing a sin x + b cos x as R sin(x + alpha) to find its maximum and minimum, and bounding expressions like a sin^2 x + b cos^2 x or sin x cos x.

Figure 1
Graphs. sin⁡x\sin x and cos⁡x\cos x are smooth waves with period 2π2\pi and range [−1,1][-1, 1]; cos⁡x\cos x is sin⁡x\sin x shifted left by π2\frac{\pi}{2}. tan⁡x\tan x has period π\pi, range R\mathbb{R}, and vertical asymptotes at x=(2n+1)π2x = (2n + 1)\frac{\pi}{2}, where it is undefined; it increases on each interval between asymptotes. cot⁡x\cot x has period π\pi and asymptotes at x=nπx = n\pi. sec⁡x\sec x and csc⁡x\csc x have period 2π2\pi and range (−∞,−1]∪[1,∞)(-\infty, -1] \cup [1, \infty). The figure shows sin⁡\sin, cos⁡\cos and tan⁡\tan over one period. Periods. sin⁡(ax+b)\sin(ax + b) and cos⁡(ax+b)\cos(ax + b) have period 2π∣a∣\frac{2\pi}{|a|}; tan⁡(ax+b)\tan(ax + b) has period π∣a∣\frac{\pi}{|a|}. Even powers and absolute values halve the period: sin⁡2x\sin^{2}x, ∣sin⁡x∣|\sin x|, ∣cos⁡x∣|\cos x| have period π\pi; sin⁡4x+cos⁡4x\sin^{4}x + \cos^{4}x and ∣sin⁡x∣+∣cos⁡x∣|\sin x| + |\cos x| have period π2\frac{\pi}{2}. For sums, the LCM of the periods is a period, but check smaller candidates (see the Functions chapter). The expression asin⁡x+bcos⁡xa\sin x + b\cos x. Write a=Rcos⁡αa = R\cos\alpha and b=Rsin⁡αb = R\sin\alpha with R=a2+b2R = \sqrt{a^{2} + b^{2}}. Then asin⁡x+bcos⁡x=Rsin⁡(x+α)a\sin x + b\cos x = R\sin(x + \alpha). Since sin⁡\sin ranges over [−1,1][-1, 1], the expression ranges over [−a2+b2,a2+b2][-\sqrt{a^{2} + b^{2}}, \sqrt{a^{2} + b^{2}}], with the maximum where x+α=π2x + \alpha = \frac{\pi}{2}. Geometrically, asin⁡x+bcos⁡xa\sin x + b\cos x is the dot product of (a,b)(a, b) with the unit vector (sin⁡x,cos⁡x)(\sin x, \cos x), which is at most ∣(a,b)∣|(a, b)| by Cauchy-Schwarz. Consequently, asin⁡x+bcos⁡x+ca\sin x + b\cos x + c has range [c−a2+b2,c+a2+b2][c - \sqrt{a^{2} + b^{2}}, c + \sqrt{a^{2} + b^{2}}], and the equation asin⁡x+bcos⁡x=ca\sin x + b\cos x = c has real solutions iff ∣c∣≤a2+b2|c| \le \sqrt{a^{2} + b^{2}}. Other standard bounds. sin⁡xcos⁡x=12sin⁡2x∈[−12,12]\sin x\cos x = \frac{1}{2}\sin 2x \in [-\frac{1}{2}, \frac{1}{2}]. asin⁡2x+bcos⁡2xa\sin^{2}x + b\cos^{2}x lies between aa and bb (it is a weighted average of aa and bb with weights sin⁡2x\sin^{2}x and cos⁡2x\cos^{2}x). a2sec⁡2x+b2csc⁡2x≥(a+b)2a^{2}\sec^{2}x + b^{2}\csc^{2}x \ge (a + b)^{2} (expand with sec⁡2=1+tan⁡2\sec^{2} = 1 + \tan^{2} and use AM-GM on a2tan⁡2x+b2cot⁡2x≥2aba^{2}\tan^{2}x + b^{2}\cot^{2}x \ge 2ab). sin⁡2x+csc⁡2x≥2\sin^{2}x + \csc^{2}x \ge 2 with equality when sin⁡2x=1\sin^{2}x = 1. Quadratic expressions in sin⁡x\sin x or cos⁡x\cos x: substitute t=sin⁡x∈[−1,1]t = \sin x \in [-1, 1] and find the range of the quadratic on that interval (the vertex may or may not lie inside). Graphs for counting solutions. The number of solutions of an equation like sin⁡x=x10\sin x = \frac{x}{10} is the number of intersections of the two graphs; count by sketching over the range where the line lies within [−1,1][-1, 1]. Key Takeaways: - sin⁡,cos⁡\sin, \cos: period 2π2\pi; tan⁡,cot⁡\tan, \cot: period π\pi; sin⁡(ax+b)\sin(ax + b): period 2π∣a∣\frac{2\pi}{|a|}. - asin⁡x+bcos⁡x=a2+b2sin⁡(x+α)a\sin x + b\cos x = \sqrt{a^{2} + b^{2}}\sin(x + \alpha); range [−a2+b2,a2+b2][-\sqrt{a^{2} + b^{2}}, \sqrt{a^{2} + b^{2}}]. - asin⁡x+bcos⁡x=ca\sin x + b\cos x = c solvable iff ∣c∣≤a2+b2|c| \le \sqrt{a^{2} + b^{2}}. - asin⁡2x+bcos⁡2xa\sin^{2}x + b\cos^{2}x lies between aa and bb. - Quadratics in sin⁡x\sin x: substitute t∈[−1,1]t \in [-1, 1] and check the vertex. - ∣sin⁡x∣|\sin x|, sin⁡2x\sin^{2}x: period π\pi. - Pitfall: taking the vertex value of a quadratic in sin⁡x\sin x when the vertex lies outside [−1,1][-1, 1]. Brain-Teaser 1: Find the range of f(x)=3sin⁡x−4cos⁡x+7f(x) = 3\sin x - 4\cos x + 7. Solution: R=9+16=5R = \sqrt{9 + 16} = 5, so 3sin⁡x−4cos⁡x∈[−5,5]3\sin x - 4\cos x \in [-5, 5] and f(x)∈[2,12]f(x) \in [2, 12]. The maximum occurs where sin⁡(x−α)=1\sin(x - \alpha) = 1 with tan⁡α=43\tan\alpha = \frac{4}{3}. No calculus is needed. Brain-Teaser 2: Find the range of g(x)=cos⁡2x+3sin⁡xg(x) = \cos 2x + 3\sin x. Solution: cos⁡2x=1−2sin⁡2x\cos 2x = 1 - 2\sin^{2}x, so with t=sin⁡x∈[−1,1]t = \sin x \in [-1, 1]: g=−2t2+3t+1g = -2t^{2} + 3t + 1. The vertex is at t=34t = \frac{3}{4}, inside [−1,1][-1, 1], giving the maximum −98+94+1=178-\frac{9}{8} + \frac{9}{4} + 1 = \frac{17}{8}. The minimum is at an endpoint: g(−1)=−2−3+1=−4g(-1) = -2 - 3 + 1 = -4 and g(1)=2g(1) = 2, so the minimum is −4-4. Range [−4,178][-4, \frac{17}{8}]. Brain-Teaser 3: How many solutions does sin⁡x=x10\sin x = \frac{x}{10} have? Solution: Solutions can only occur where ∣x10∣≤1|\frac{x}{10}| \le 1, i.e. ∣x∣≤10|x| \le 10. Both sides are odd, so count x>0x > 0 and double, then add x=0x = 0. On (0,10](0, 10]: the line rises from 00 to 11 while sin⁡x\sin x completes about 1.61.6 periods (10≈3.18π10 \approx 3.18\pi). The positive humps of sin⁡x\sin x are on (0,π)(0, \pi) and (2π,3π)(2\pi, 3\pi), and each crosses the line twice, except that on (0,π)(0, \pi) one crossing is at the origin itself; so there is 11 positive solution in (0,π)(0, \pi) and 22 in (2π,3π)(2\pi, 3\pi), giving 33 positive solutions. By symmetry 33 negative, plus x=0x = 0: 77 solutions in total.

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Graphs of Trigonometric Functions, Periods and the Range of a sin x + b cos x — DPP

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