Exercises 16 (Trigonometric functions and equations)

  1. Determine all the angles between 0 and 360 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