Question

Evaluate these trigonometric expressions involving sine, cosine and tangent functions. Solve the following for 0° ≤  ≤ 360°

a) 2 sin²  − 3 sin  = 0

b) 2 cos³  − cos  = 0

c) tan²  − tan  − 1 = 0

Answer 1

(i) \(2sin^2x - 3sinx = 0\)

⇒ \(sinx(2sinx -3) = 0\)

⇒ \(sinx = 0 \;or\; sinx = \frac 32 \) ⇒ \(x = sin^{-1}(\frac 32)\)

⇒ \(x = n\pi \;or\;x = n\pi + (-1)^n sin^{-1} (\frac 32)\)

(ii) \(2cos^3x - cosx = 0\)

⇒ \(cosx(2cos^2x - 1) = 0\)

⇒ \(cosx.cos2x = 0\)

⇒ \(cos x = 0\;or\;cos 2x = 0\)

⇒ \(x = 2n\pi \pm \frac \pi2 \;or\; x = n\pi \pm \frac \pi4\)

(iii) \(tan^2x - tanx -1 =0\)

⇒ \(tanx = \frac{1 \pm \sqrt 5}2\)

⇒ \(x = -9.5° \; or\; 9.5° \)  (approx)

⇒ \(x = 180° n\pm 9.5° \)