# 20.3: Exercises

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## 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. $$\cos(x)=0.2$$
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. $$\sin(x)=-0.06$$
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. $$\sin(x)=-1$$
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. $$\tan^3(x)-\tan(x)=0$$
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. $$\sin(x)+\tan(x)=\cos(x)$$
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. $$x \approx 0.443+2 n \pi, \text { or } x \approx 2.193+2 n \pi, \text { where } n=0, \pm 1, \ldots$$