20.3: Exercises

Exercise $\PageIndex{1}$

Find all solutions of the equation, and simplify as much as possible. Do not approximate the solution.

1. $\tan(x)=\dfrac{\sqrt{3}}{3}$
2. $\sin(x)=\dfrac{\sqrt{3}}{2}$
3. $\sin(x)=-\dfrac{\sqrt{2}}{2}$
4. $\cos(x)=\dfrac{\sqrt{3}}{2}$
5. $\cos(x)=0$
6. $\cos(x)=-0.5$
7. $\cos(x)=1$
8. $\sin(x)=5$
9. $\sin(x)=0$
10. $\sin(x)=-1$
11. $\tan(x)=-\sqrt{3}$
12. $\cos(x)=0.2$

Answer

1. $x=\dfrac{\pi}{6}+n \pi \text {, where } n=0, \pm 1, \ldots$
2. $x=(-1)^{n} \dfrac{\pi}{3}+n \pi \text {, where } n=0, \pm 1, \ldots$
3. $x=(-1)^{n+1} \dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots$
4. $x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
5. $x=\pm \dfrac{\pi}{2}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
6. $x=\pm \dfrac{2 \pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
7. $x=2 n \pi, \text { where } n=0, \pm 1, \ldots$
8. there is no solution (since $-1 \leq \sin (x) \leq$),
9. $x=n \pi, \text { where } n=0, \pm 1, \ldots$
10. $x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots$ (since each solution appears twice, it is enough to take $n=0, \pm 2, \pm 4, \ldots$),
11. $x=\dfrac{-\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
12. $x=\pm \cos ^{-1}(0.2)+2 n \pi, \text { where } n=0, \pm 1, \ldots$

Exercise $\PageIndex{2}$

Find all solutions of the equation. Approximate your solution with the calculator.

1. $\tan(x)=6.2$
2. $\cos(x)=0.45$
3. $\sin(x)=0.91$
4. $\cos(x)=-.772$
5. $\tan(x)=-0.2$
6. $\sin(x)=-0.06$

Answer

1. $x \approx 1.411+n \pi, \text { where } n=0, \pm 1, \ldots$
2. $x \approx \pm 1.104+2 n \pi, \text { where } n=0, \pm 1, \ldots$
3. $x \approx(-1)^{n} 1.143+n \pi, \text { where } n=0, \pm 1, \ldots$
4. $x \approx \pm 2.453+2 n \pi, \text { where } n=0, \pm 1, \ldots$
5. $x \approx-0.197+n \pi, \text { where } n=0, \pm 1, \ldots$
6. $x \approx(-1)^{n+1} 0.06+n \pi, \text { where } n=0, \pm 1, \ldots$

Exercise $\PageIndex{3}$

Find at least $5$ distinct solutions of the equation.

1. $\tan(x)=-1$
2. $\cos(x)=\dfrac{\sqrt{2}}{2}$
3. $\sin(x)=-\dfrac{\sqrt{3}}{2}$
4. $\tan(x)=0$
5. $\cos(x)=0$
6. $\cos(x)=0.3$
7. $\sin(x)=0.4$
8. $\sin(x)=-1$

Answer

1. $\dfrac{-\pi}{4}, \dfrac{3 \pi}{4}, \dfrac{7 \pi}{4}, \dfrac{-5 \pi}{4}, \dfrac{-9 \pi}{4}$
2. $\dfrac{\pi}{4}, \dfrac{-\pi}{4}, \dfrac{9 \pi}{4}, \dfrac{-9 \pi}{4}, \dfrac{17 \pi}{4}, \dfrac{-17 \pi}{4}$
3. $\dfrac{-\pi}{3}, \dfrac{4 \pi}{3}, \dfrac{5 \pi}{3}, \dfrac{-2 \pi}{3}, \dfrac{-7 \pi}{3}$
4. $0, \pi, 2 \pi,-\pi,-2 \pi,$
5. $\dfrac{\pi}{2}, \dfrac{-\pi}{2}, \dfrac{3 \pi}{2}, \dfrac{-3 \pi}{2}, \dfrac{5 \pi}{2}, \dfrac{-5 \pi}{2}$
6. $\cos ^{-1}(0.3),-\cos ^{-1}(0.3), \cos ^{-1}(0.3)+2 \pi,-\cos ^{-1}(0.3)+2 \pi, \cos ^{-1}(0.3)-2 \pi,-\cos ^{-1}(0.3)-2 \pi$
7. $\sin ^{-1}(0.4),-\sin ^{-1}(0.4)+\pi, -\sin ^{-1}(0.4)-\pi, \sin ^{-1}(0.4)+2 \pi, \sin ^{-1}(0.4)-2 \pi$
8. $\dfrac{3 \pi}{2}, \dfrac{7 \pi}{2}, \dfrac{11 \pi}{2}, \dfrac{-\pi}{2}, \dfrac{-5 \pi}{2}$

Exercise $\PageIndex{4}$

Solve for $x$. State the general solution without approximation.

1. $\tan(x)-1=0$
2. $2\sin(x)=1$
3. $2\cos(x)-\sqrt{3}=0$
4. $\sec(x)=-2$
5. $\cot(x)=\sqrt{3}$
6. $\tan^2(x)-3=0$
7. $\sin^2(x)-1=0$
8. $\cos^2(x)+7\cos(x)+6=0$
9. $4\cos^2(x)-4\cos(x)+1=0$
10. $2\sin^2(x)+11\sin(x)=-5$
11. $2\sin^2(x)+\sin(x)-1=0$
12. $2\cos^2(x)-3\cos(x)+1=0$
13. $2\cos^2(x)+9\cos(x)=5$
14. $\tan^3(x)-\tan(x)=0$

Answer

1. $x=\dfrac{\pi}{4}+n \pi, \text { where } n=0, \pm 1, \ldots$
2. $x=(-1)^{n} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
3. $x=\pm \dfrac{\pi}{6}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
4. $x=\pm \dfrac{2 \pi}{3}+2 n \pi \text { where } n=0, \pm 1, \ldots$
5. $x=\dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
6. $x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
7. $x=\pm \dfrac{\pi}{2}+n \pi, \text { where } n=0, \pm 1, \ldots$
8. $x=\pi+2 n \pi \text {, where } n=0, \pm 1, \ldots$ (Note: The solution given by the formula 20.1.5 is $x=\pm \pi+2 n \pi \text { with } n=0, \pm 1, \ldots$ Since every solution appears twice in this expression, we can reduce this to $x=\pi+2 n \pi$.),
9. $x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
10. $x=(-1)^{n+1} \dfrac{\pi}{6}+n \pi, \text { where } n=0, \pm 1, \ldots$
11. $x=(-1)^{n+1} \dfrac{\pi}{2}+n \pi$
12. $x=2 n \pi, \text { or } x=\pm \dfrac{\pi}{3}+2 n \pi, \text { where } n=0, \pm 1, \ldots$
13. $x=\pm \dfrac{\pi}{3}+n \pi, \text { where } n=0, \pm 1, \ldots$
14. $x=\pm \dfrac{\pi}{4}+n \pi, \text { or } x=n \pi, \text { where } n=0, \pm 1, \ldots$

Exercise $\PageIndex{5}$

Use the calculator to find all solutions of the given equation. Approximate the answer to the nearest thousandth.

1. $2\cos(x)=2\sin(x)+1$
2. $7\tan(x)\cdot \cos(2x)=1$
3. $4\cos^2(3x)+\cos(3x)=\sin(3x)+2$
4. $\sin(x)+\tan(x)=\cos(x)$

Answer

1. $x \approx-1.995+2 n \pi, \text { or } x \approx 0.424+2 n \pi, \text { where } n=0, \pm 1, \ldots$
2. $x \approx-0.848+n \pi, \text { or } x \approx 0.148+n \pi, \text { or } x \approx 0.700+n \pi, \text { where } n=0, \pm 1, \ldots$
3. $x \approx 0.262+n \dfrac{2 \pi}{3} \text {, or } x \approx 0.906+n \dfrac{2 \pi}{3} \text {, or } x \approx 1.309+n \dfrac{2 \pi}{3}, \text { or } x \approx 1.712+n \dfrac{2 \pi}{3}, \text { where } n=0, \pm 1, \ldots$
4. $x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots$