Trigonometry(Part 3)

Trigonometry (Part 3)

6.16 Solution of trigonometric equations

For questions 1 to 8, solve the equations for $0^{\circ} \le x \le 360^{\circ}$, giving your answers correct to one decimal place

1. a. $\sin (x+20^{\circ})=0.4$

b. $\cos (x-50^{\circ})=-0.3$

2. $\sin 2x=6\cos x$

3. $3\sin 2x=2\sin x$

4. $2\sec ^{2}x-1=3\tan x$

5. $2\cos 2x -5\cos x+3=0$

6. $7-3\cos 2x-10\sin x=0$

7. a. $3\sin 2\theta =\sin \theta$

b. $4\cos \theta = 3\sin 2\theta$

c. $\sin 2\theta + \cos \theta = 0$

d. $3\cos 2\theta - \cos \theta +2=0$

8. a. $3\sqrt{\cot (\theta -10^{\circ})}=4$

b. $4\cot ^{2}\theta +6=11\cot \theta$

c. $5\sec ^{2}\theta – 23\sec \theta =10$

d. $9\sec ^{2}\theta = 24\sec \theta -16$

e. $8$cosec$^{2}\theta=14$cosec$\theta - 5$

f. $2\sec ^{2}\theta - \tan \theta -3=0$

g. $2$cosec$^{2}\theta +\cot \theta =8$

h. $5\tan \theta +6=2\sec ^{2}\theta$

i. $3\cos \theta +3=\frac{5}{cosec ^{2}\theta}$

j. $2\sin \theta +1=\frac{3}{cosec^{2}\theta}$

k. $3\sec ^{2}\theta = 7+\frac{5}{\cot \theta}$

9. Given that cosec$C=7$, $\sin ^{2}D=\frac{1}{2}$, and $\tan ^{2}E=4$, find the possible values of $\cot C$, $\sec D$ and cosec$E$, giving your answers in exact form.

10. Solve these equations for values of $A$ between $0$ and $2\pi$ inclusive.

     a. $\cos 2A+3+4\cos A=0$

     b. $2\cos 2A+1+\sin A=0$

6.17 More exercise

1. Prove the identity $\cot \theta - \tan \theta \equiv 2\cot 2\theta$

2. i) Express $4\sin \theta – 3\cos \theta$ in the form $R\sin (\theta - \alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$, stating the value $\alpha$ correct to $2$ decimal places.

Hence,

ii) solve the equation $4\sin \theta – 3\cos \theta =2$, giving all values of $\theta$ such that $0^{\circ}<\theta < 360^{\circ}$

iii) write down the greatest value of $\frac{1}{4\sin \theta – 3\cos \theta +6}$

3. i) Show that the equation $\sin (x-60^{\circ})-\cos (30^{\circ}-x)=1$ can be written in the form $\cos x=k$, where $k$ is a constant.

ii) Hence solve the equation, for $0^{\circ}< x <180^{\circ}$

4. Prove the identity $\cot x - \cot 2x \equiv$cosec$2x$

5. Solve the equation $\cos \theta + 3\cos 2\theta = 2$, giving all solutions in the interval $0^{\circ} \le \theta \le 180^{\circ}$

6. Sketch the graph of $y=\sec x$, for $0\le x \le 2\pi$

7. Prove the identity $\sin ^{2}\theta \cos ^{2}\theta \equiv \frac{1}{8}(1-\cos 4\theta)$

8. i) Show that the equation $\tan (45^{\circ}+x)=2\tan (45^{\circ}-x)$ can be written in the form $\tan ^{2}x-6\tan x+1=0$

ii) Hence solve the equation $\tan (45^{\circ}+x)=2\tan (45^{\circ}-x)$, for $0^{\circ}< x < 90^{\circ}$

9. i) Prove the identity $\cos 4\theta + 4\cos 2\theta \equiv 8\cos ^{4}\theta -3$

ii) Hence solve the equation $\cos 4\theta + 4\cos 2\theta =2$, for $0^{\circ}\le \theta \le 360^{\circ}$

10. By expressing $8\sin \theta – 6\cos \theta$ in the form $R\sin (\theta - \alpha)$, solve the equation $8\sin \theta – 6\cos \theta =7$, for $0^{\circ}\le \theta \le 360^{\circ}$.

11. i) Express $7\cos \theta +24\sin \theta$ in the form $R\cos (\theta - \alpha)$, where $R>0$ and $0^{\circ}< \alpha < 90^{\circ}$, giving the exact value of $R$ and the value of $\alpha$ correct to $2$ decimal places

  ii) Hence solve the equation $7\cos \theta +24\sin \theta =15$, giving all solutions in the interval $0^{\circ}\le \theta \le 360^{\circ}$.

12. Solve the equation $\tan x\tan 2x=1$, giving all solutions in the interval $0^{\circ}< x < 180^{\circ}$

13. Express $\cos \theta +(\sqrt{3})\sin \theta$ in the form $R\cos (\theta - \alpha)$, where $R>0$ and $0 < \alpha < \frac{1}{2}\pi$, giving the exact values of $R$ and $\alpha$

14. i) Show that the equation $\tan (45^{\circ}+x)-\tan x=2$ can be written in the form $\tan ^{2}x+2\tan x-1=0$

  ii) Hence solve the equation $\tan (45^{\circ}+x)-\tan x=2$, giving all solutions in the interval $0^{\circ}\le \theta \le 180^{\circ}$.

15. i) Show that the equation $\tan (30^{\circ}+\theta)=2\tan (60^{\circ}-\theta)$ can be written in the form $\tan ^{2}\theta +(6\sqrt{3})\tan \theta – 5=0$

  ii) Hence, or otherwise, solve the equation $\tan (30^{\circ}+\theta)=2\tan (60^{\circ}-\theta)$, for $0^{\circ}\le \theta \le 180^{\circ}$.

16. i) Express $5\sin x+12\cos x$ in the form $R\sin (x+\alpha)$, where $R>0$ and $0^{\circ}< \alpha <90^{\circ}$, giving the value of $\alpha$ correct to $2$ decimal places

      ii) Hence solve the equation $5\sin 2\theta + 12\cos 2\theta = 11$, giving all solutions in the interval $0^{\circ}< \theta < 180^{\circ}$