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) \(\pm45^\circ\), \(\pm135^\circ\)