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Section 14.3 Introduction to Solving Trigonometric Equations
Subsection 14.3.1 Basic Trigonometric Equations
Exercise Group 14.3.1 . Solving Basic Equations.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\) and the general solution.
(a) \(\sin{x} = \frac{1}{2}\) Answer .
\(x = \frac{\pi}{6}, \frac{5\pi}{6}\text{.}\) General solution:
\(x = \frac{\pi}{6} + 2\pi n, x = \frac{5\pi}{6} + 2\pi n, n \in \mathbb{I}\text{.}\)
(b) \(\cos{x} = -\frac{1}{\sqrt{2}}\) Answer .
\(x = \frac{3\pi}{4}, \frac{5\pi}{4}\text{.}\) General solution:
\(x = \frac{3\pi}{4} + 2\pi n, x = \frac{5\pi}{4} + 2\pi n, n \in \mathbb{I}\text{.}\)
(c) \(\sin{x} = -\frac{\sqrt{3}}{2}\) Answer .
\(x = \frac{4\pi}{3}, \frac{5\pi}{3}\text{.}\) General solution:
\(x = \frac{4\pi}{3} + 2\pi n, x = \frac{5\pi}{3} + 2\pi n, n \in \mathbb{I}\text{.}\)
(d) \(\cos{x} = 1\) Answer .
\(x = 0\text{.}\) General solution:
\(x = 2\pi n, n \in \mathbb{I}\text{.}\)
(e) \(\sin{x} = \frac{\sqrt{3}}{2}\) Answer .
\(x = \frac{\pi}{3}, \frac{2\pi}{3}\text{.}\) General solution:
\(x = \frac{\pi}{3} + 2\pi n, x = \frac{2\pi}{3} + 2\pi n, n \in \mathbb{I}\text{.}\)
(f) \(\sin{x} = 1\) Answer .
\(x = \frac{\pi}{2}\text{.}\) General solution:
\(x = \frac{\pi}{2} + 2\pi n, n \in \mathbb{I}\text{.}\)
(g) \(\tan{x} = 0\) Answer .
\(x = 0, \pi\text{.}\) General solution:
\(x = \pi n, n \in \mathbb{I}\text{.}\)
(h) \(\sin{x} = 0\) Answer .
\(x = 0, \pi\text{.}\) General solution:
\(x = \pi n, n \in \mathbb{I}\text{.}\)
(i) \(\tan{x} = \sqrt{3}\) Answer .
\(x = \frac{\pi}{3}, \frac{4\pi}{3}\text{.}\) General solution:
\(x = \frac{\pi}{3} + \pi n, n \in \mathbb{I}\text{.}\)
(j) \(\sin{x} = -\frac{1}{2}\) Answer .
\(x = \frac{7\pi}{6}, \frac{11\pi}{6}\text{.}\) General solution:
\(x = \frac{7\pi}{6} + 2\pi n, x = \frac{11\pi}{6} + 2\pi n, n \in \mathbb{I}\text{.}\)
(k) \(\cos{x} = \frac{1}{2}\) Answer .
\(x = \frac{\pi}{3}, \frac{5\pi}{3}\text{.}\) General solution:
\(x = \frac{\pi}{3} + 2\pi n, x = \frac{5\pi}{3} + 2\pi n, n \in \mathbb{I}\text{.}\)
(l) \(\sin{x} = -1\) Answer .
\(x = \frac{3\pi}{2}\text{.}\) General solution:
\(x = \frac{3\pi}{2} + 2\pi n, n \in \mathbb{I}\text{.}\)
(m) \(\cos{x} = -\frac{1}{2}\) Answer .
\(x = \frac{2\pi}{3}, \frac{4\pi}{3}\text{.}\) General solution:
\(x = \frac{2\pi}{3} + 2\pi n, x = \frac{4\pi}{3} + 2\pi n, n \in \mathbb{I}\text{.}\)
(n) \(\cos{x} = -1\) Answer .
\(x = \pi\text{.}\) General solution:
\(x = \pi + 2\pi n, n \in \mathbb{I}\text{.}\)
(o) \(\cos{x} = \frac{\sqrt{3}}{2}\) Answer .
\(x = \frac{\pi}{6}, \frac{11\pi}{6}\text{.}\) General solution:
\(x = \frac{\pi}{6} + 2\pi n, x = \frac{11\pi}{6} + 2\pi n, n \in \mathbb{I}\text{.}\)
(p) \(\tan{x} = 1\) Answer .
\(x = \frac{\pi}{4}, \frac{5\pi}{4}\text{.}\) General solution:
\(x = \frac{\pi}{4} + \pi n, n \in \mathbb{I}\text{.}\)
(q) \(\sin{x} = \frac{1}{\sqrt{2}}\) Answer .
\(x = \frac{\pi}{4}, \frac{3\pi}{4}\text{.}\) General solution:
\(x = \frac{\pi}{4} + 2\pi n, x = \frac{3\pi}{4} + 2\pi n, n \in \mathbb{I}\text{.}\)
Subsection 14.3.2 Examples
Exercise Group 14.3.2 . Isolating Trigonometric Functions.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\) and the general solution.
(a) \(2\sin{x}=-1\) Answer .
\(x=\frac{7\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{7\pi}{6}+2\pi n,x=\frac{11\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(b) \(\cos{x}-1=-\cos{x}\) Answer .
\(x=\frac{\pi}{3},\frac{5\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{3}+2\pi n,x=\frac{5\pi}{3}+2\pi n,n \in \mathbb{I}\text{.}\)
(c) \(3\sin{x}=\sin{x}+1\) Answer .
\(x=\frac{\pi}{6},\frac{5\pi}{6}\text{.}\) General solution:
\(x=\frac{\pi}{6}+2\pi n,x=\frac{5\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(d) \(5\cos{x}-\sqrt{3}=3\cos{x}\) Answer .
\(x=\frac{\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{\pi}{6}+2\pi n,x=\frac{11\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(e) \(5\sin{x}+1=3\sin{x}\) Answer .
\(x=\frac{7\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{7\pi}{6}+2\pi n,x=\frac{11\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(f) \(5\sin{x}+1=6\) Answer .
\(x=\frac{\pi}{2}\text{.}\) General solution:
\(x=\frac{\pi}{2}+2\pi n,n \in \mathbb{I}\text{.}\)
(g) \(5\tan{x}+5=0\) Answer .
\(x=\frac{3\pi}{4},\frac{7\pi}{4}\text{.}\) General solution:
\(x=\frac{3\pi}{4}+\pi n,n \in \mathbb{I}\text{.}\)
(h) \(6\sin{x}-8=2\sqrt{3}+2\sin{x}-8\) Answer .
\(x=\frac{\pi}{3},\frac{2\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{3}+2\pi n,x=\frac{2\pi}{3}+2\pi n,n \in \mathbb{I}\text{.}\)
(i) \(\sin{x}=\sqrt{3}-\sin{x}\) Answer .
\(x=\frac{\pi}{3},\frac{2\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{3}+2\pi n,x=\frac{2\pi}{3}+2\pi n,n \in \mathbb{I}\text{.}\)
(j) \(\sqrt{3}\tan{x}=-1\) Answer .
\(x=\frac{5\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{5\pi}{6}+\pi n,n \in \mathbb{I}\text{.}\)
(k) \(\cos{x}=\sqrt{3}-\cos{x}\) Answer .
\(x=\frac{\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{\pi}{6}+2\pi n,x=\frac{11\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(l) \(5(1+2\sin{x})=2\sin{x}+1\) Answer .
\(x=\frac{7\pi}{6},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{7\pi}{6}+2\pi n,x=\frac{11\pi}{6}+2\pi n,n \in \mathbb{I}\text{.}\)
(m) \(\sqrt{3}\tan{x}+5=6\) Answer .
\(x=\frac{\pi}{6},\frac{7\pi}{6}\text{.}\) General solution:
\(x=\frac{\pi}{6}+\pi n,n \in \mathbb{I}\text{.}\)
(n) \(3\tan{x}-3=5\tan{x}-1\) Answer .
\(x=\frac{3\pi}{4},\frac{7\pi}{4}\text{.}\) General solution:
\(x=\frac{3\pi}{4}+\pi n,n \in \mathbb{I}\text{.}\)
(o) \(2-2\cot{x}=0\) Answer .
\(x=\frac{\pi}{4},\frac{5\pi}{4}\text{.}\) General solution:
\(x=\frac{\pi}{4}+\pi n,n \in \mathbb{I}\text{.}\)
If the ratio is not of a special angle, then you need to use your calculator’s inverse trig functions to find the reference angle.
Exercise Group 14.3.3 . Using Inverse Trigonometric Functions.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\) and the general solution.
(a) \(\sin{x}=0.159\) Answer .
\(x\approx0.16,2.98\text{.}\) General solution:
\(x\approx0.16+2\pi n, x\approx2.98+2\pi n, n \in \mathbb{I}\text{.}\)
(b) \(\cos{x}=-0.513\) Answer .
\(x\approx2.11,4.17\text{.}\) General solution:
\(x\approx2.11+2\pi n, x\approx4.17+2\pi n, n \in \mathbb{I}\text{.}\)
(c) \(\tan{x}=5.23\) Answer .
\(x\approx1.38,4.52\text{.}\) General solution:
\(x\approx1.38+\pi n, n \in \mathbb{I}\text{.}\)
(d) \(\sin{x}=-0.373\) Answer .
\(x\approx3.52,5.90\text{.}\) General solution:
\(x\approx3.52+2\pi n, x\approx5.90+2\pi n, n \in \mathbb{I}\text{.}\)
(e) \(\cos{x}=0.276\) Answer .
\(x\approx1.29,4.99\text{.}\) General solution:
\(x\approx1.29+2\pi n, x\approx4.99+2\pi n, n \in \mathbb{I}\text{.}\)
(f) \(\tan{x}=-0.618\) Answer .
\(x\approx2.59,5.73\text{.}\) General solution:
\(x\approx2.59+\pi n, n \in \mathbb{I}\text{.}\)
(g) \(\cos{x}=\frac{1}{3}\) Answer .
\(x\approx1.23,5.05\text{.}\) General solution:
\(x\approx1.23+2\pi n, x\approx5.05+2\pi n, n \in \mathbb{I}\text{.}\)
(h) \(\sin{x}=-\frac{2}{3}\) Answer .
\(x\approx3.87,5.55\text{.}\) General solution:
\(x\approx3.87+2\pi n, x\approx5.55+2\pi n, n \in \mathbb{I}\text{.}\)
(i) \(\tan{x}=-4.87\) Answer .
\(x\approx1.77,4.91\text{.}\) General solution:
\(x\approx1.77+\pi n, n \in \mathbb{I}\text{.}\)
(j) \(\tan{x}=4.36\) Answer .
\(x\approx1.35,4.49\text{.}\) General solution:
\(x\approx1.35+\pi n, n \in \mathbb{I}\text{.}\)
(k) \(\cos{x}=-0.19\) Answer .
\(x\approx1.76,4.52\text{.}\) General solution:
\(x\approx1.76+2\pi n, x\approx4.52+2\pi n, n \in \mathbb{I}\text{.}\)
Exercise Group 14.3.4 . Solving in a Specific Interval.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\text{.}\)
(a) \(3\cos{x}=-2\)
(b) \(2\tan{x}=3\)
(c) \(-3\sin{x}-1=1\)
(d) \(3\tan{x}+5=0\)
(e) \(3(\tan{x}+1)=2\)
(f) \(-5\cos{x}+3=2\)
(g) \(8-\tan{x}=10\)
Subsection 14.3.3 Solving Equations with a Multiple of the Angle
Exercise Group 14.3.5 . Equations with Multiple Angles.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\) and the general solution.
(a) \(\sin{(2x)}=\frac{\sqrt{3}}{2}\) Answer .
\(x=\frac{\pi}{6},\frac{\pi}{3},\frac{7\pi}{6},\frac{4\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{6}+\pi n, x=\frac{\pi}{3}+\pi n, n \in \mathbb{I}\text{.}\)
(b) \(\tan{(3x)}=-1\) Answer .
\(x=\frac{\pi}{4},\frac{7\pi}{12},\frac{11\pi}{12},\frac{5\pi}{4},\frac{19\pi}{12},\frac{23\pi}{12}\text{.}\) General solution:
\(x=\frac{\pi}{4}+\frac{2\pi}{3}n, x=\frac{7\pi}{12}+\frac{2\pi}{3}n, n \in \mathbb{I}\text{.}\)
(c) \(\cos{(2x)}=-\frac{1}{2}\) Answer .
\(x=\frac{\pi}{3},\frac{2\pi}{3},\frac{4\pi}{3},\frac{5\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{3}+\pi n, x=\frac{2\pi}{3}+\pi n, n \in \mathbb{I}\text{.}\)
(d) \(\tan{(4x)}=\sqrt{3}\) Answer .
\(x=\frac{\pi}{12},\frac{\pi}{3},\frac{7\pi}{12},\frac{5\pi}{6},\frac{13\pi}{12},\frac{4\pi}{3},\frac{19\pi}{12},\frac{11\pi}{6}\text{.}\) General solution:
\(x=\frac{\pi}{12}+\frac{\pi}{4}n, n \in \mathbb{I}\text{.}\)
(e) \(\cos{(2x)}=0\) Answer .
\(x=\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},\frac{7\pi}{4}\text{.}\) General solution:
\(x=\frac{\pi}{4}+\frac{\pi}{2}n, n \in \mathbb{I}\text{.}\)
(f) \(\cos{(4x)}=\frac{\sqrt{2}}{2}\) Answer .
\(x=\frac{\pi}{16},\frac{7\pi}{16},\frac{9\pi}{16},\frac{15\pi}{16},\frac{17\pi}{16},\frac{23\pi}{16},\frac{25\pi}{16},\frac{31\pi}{16}\text{.}\) General solution:
\(x=\frac{\pi}{16}+\frac{\pi}{2}n, x=\frac{7\pi}{16}+\frac{\pi}{2}n, n \in \mathbb{I}\text{.}\)
(g) \(\sin{(3x)}=0\) Answer .
\(x=0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{5\pi}{3}\text{.}\) General solution:
\(x=\frac{\pi}{3}n, n \in \mathbb{I}\text{.}\)
(h) \(\sin{(2x)}=-\frac{1}{2}\) Answer .
\(x=\frac{7\pi}{12},\frac{11\pi}{12},\frac{19\pi}{12},\frac{23\pi}{12}\text{.}\) General solution:
\(x=\frac{7\pi}{12}+\pi n, x=\frac{11\pi}{12}+\pi n, n \in \mathbb{I}\text{.}\)
(i) \(\tan{\left(\frac{1}{2}x\right)}=-\sqrt{3}\) Answer .
\(x=\frac{4\pi}{3}\text{.}\) General solution:
\(x=\frac{4\pi}{3}+2\pi n, n \in \mathbb{I}\text{.}\)
Exercise Group 14.3.6 . Complex Arguments.
Solve each equation, by finding all solutions in the interval
\([0,2\pi)\text{.}\)
(a) \(2\cos{(2x)}-1=0\) Answer .
\(x=\frac{\pi}{6},\frac{5\pi}{6},\frac{7\pi}{6},\frac{11\pi}{6}\)
(b) \(\cos{(3x)}-1=5\cos{(3x)}+2\) Answer .
\(x\approx 0.81, 1.29, 2.90, 3.38, 5.00, 5.48\)
(c) \(3\sin{(4x)}=3-2\sin{(4x)}\) Answer .
\(x\approx 0.16, 0.62, 1.73, 2.20, 3.30, 3.77, 4.87, 5.34\)
