Exercises 16 (Trigonometric functions and equations)

1. Determine all the angles between $0^\circ$ and $360^\circ$ whose

   1. sine is $-0.4848$
   2. cosine is $0.8361$
   3. tangent is $-1.832$
   4. secant is $1.392$
   5. cotangent is $0.6848$

2. Evaluate to 4 significant figures: $$\sec 286.08^\circ – 3.26 \,\mathrm{cosec}\, 146.72^\circ + 9 \cot 312.25^\circ.$$

3. Evaluate the following:

   1. $\sin\frac{3\pi}{8}$
   2. $\cos\frac{5\pi}{9}$
   3. $\tan\frac{5\pi}{16}$
   4. $\sec\frac{7\pi}{12}$
   5. $\mathrm{cosec}\,4.72\pi$
   6. $\cot \frac{4\pi}{9}$

4. Given that $A=32.9^\circ$ and $B=63.48^\circ$, determine to four significant figures:

   1. $2\sec A\cot B$
   2. $\dfrac{\mathrm{cosec}\,A+\sec B}{1-\tan A\cos B}$
   3. $\dfrac{5\cot B}{4\sin A\,\mathrm{cosec}\, B}$

5. Prove the following:

   1. $\tan^2\theta\,(\mathrm{cosec}^2\theta -1)=1$
   2. $\tan\theta = \sqrt{\dfrac{1-\cos^2\theta}{\cos^2\theta}}$
   3. $\dfrac{\mathrm{cosec}\,\theta}{\sec\theta}-\dfrac{\sec\theta}{\mathrm{cosec}\,\theta} = (\cos^2\theta-\sin^2\theta)\sec\theta\,\mathrm{cosec}\,\theta$
   4. $\sin\theta-\sin^3\theta = \dfrac{\sin\theta}{\sec^2\theta}$

6. Solve the equation $8 \sin^2\theta + 2 \cos\theta = 5$, stating all the values of $\theta$ between $0^\circ$ and $360^\circ$.

7. Solve the following equations for all values of $x$ from $0^\circ$ to $360^\circ$:

   1. $\sin x=0.3$
   2. $\cos x=-0.7$
   3. $\tan x=-0.75$
   4. $2\sin x=3\cos x$
   5. $4\sin x\cos x = 3\cos x$
   6. $4\cos^2 x+\cos x = 0$
   7. $2\sin^2 x-\sin x-1=0$
   8. $\sin x-2\cos^2 x+1=0$

8. Solve the following equations for all values of $x$ from $-180^\circ$ to $180^\circ$:

   1. $\cos^2 x=0.75$
   2. $\sin 2x = 2\cos 2x$
   3. $3\sin^2 x=2\sin x\cos x$
   4. $2\cos^2 x-5\cos x+2=0$
   5. $\sin^2 x+\cos x+1=0$
   6. $\sin^2 x+5\cos^2 x =3$

Solutions: 1. (i) $209^\circ$, $331^\circ$; (ii) $33.3^\circ$, $326.7^\circ$; (iii) $118.6^\circ$, $298.6^\circ$; (iv) $44.1^\circ$, $315.9^\circ$; (v) $55.6^\circ$, $235,6^\circ$; 2. $-10.51$; 3. (i) $0.92$; (ii) $-0.17$; (iii) $1.497$; (iv) $-3.86$ (v) $1.298$; (vi) $0.18$; 4. (i) $1.189$; (ii) $5.738$; (iii) $1.027$; 6. $41.9^\circ$, $318.6^\circ$, or $120^\circ$, $240^\circ$; 7. (a) $17.5^\circ$, $162.5^\circ$; (b) $134.4^\circ$, $225.6^\circ$; (c) $143.1^\circ$, $323.1^\circ$; (d) $56.3^\circ$, $236.3^\circ$; (e) $48.6^\circ$, $90^\circ$, $131.4^\circ$, $270^\circ$; (f) $90^\circ$, $104.5^\circ$, $255.5^\circ$, $270^\circ$; (g) $90^\circ$, $210^\circ$, $330^\circ$; (h) $30^\circ$, $150^\circ$, $270^\circ$; 8. (a) $\pm 30^\circ$, $\pm150^\circ$; (b) $-148.3^\circ$, $-58.3^\circ$, $31.7^\circ$, $121.7^\circ$; (c) $0^\circ$, $33.7^\circ$, $-146.3^\circ$, $\pm180^\circ$; (d) $\pm 60^\circ$; (e) $\pm180^\circ$; (f) $\pm45^\circ$, $\pm135^\circ$