Trigonometry
- Find the value of \(\tan 15^\circ \times \tan 30^\circ \times \tan 45^\circ \times \tan 60^\circ \times \tan 75^\circ\). Answer: A. 1 Answer: (A) 1 tanθ × tan(90^° - θ) = tanθ × cotθ = 1 . So tan 15^° tan 75^° = 1 , tan 30^° tan 60^° = 1 and tan 45^° = 1 . Prod... 1
- If \(\cos\theta = \frac{5}{13}\) and \(\theta\) is acute, find the value of \(\csc\theta + \cot\theta\). Answer: C. 3/2 Answer: (C) 3/2 With adjacent side 5 and hypotenuse 13, the opposite side is 12. cscθ = 13/12 , cotθ = 5/12 . Sum = 18/12 = 3... 1
- If \(\sec\theta + \tan\theta = 3\), find \(\sin\theta\). Answer: D. 4/5 Answer: (D) 4/5 Since sec²θ - tan²θ = 1 , secθ - tanθ = 1/3 . Adding: secθ = 5/3 ; subtracting: tanθ = 4/3 . sinθ = tanθ/secθ... 1
- Find the value of \(\frac{\sin 35^\circ}{\cos 55^\circ} + \frac{\cos 28^\circ}{\sin 62^\circ} - 2\cos^2 60^\circ\). Answer: A. 3/2 Answer: (A) 3/2 cos 55^° = sin 35^° and sin 62^° = cos 28^° (complementary angles), so the first two terms are 1 each. 2cos²... 1
- If \(\sin\theta = \frac{3}{5}\) and θ is acute, find \(\tan\theta + \sec\theta\). Answer: D. 2 Answer: (D) 2 cosθ = 4/5 , so tanθ = 3/4 , secθ = 5/4 . Sum = 3 + 5/4 = 2 . 1
- Find the value of \(\sin^2 30^\circ + \cos^2 60^\circ + \tan^2 45^\circ\). Answer: A. 3/2 Answer: (A) 3/2 sin 30^° = cos 60^° = 1/2 , tan 45^° = 1 . 1/4 + 1/4 + 1 = 3/2 . 1
- If \(\tan\theta + \cot\theta = 2\), where \(0^\circ < \theta < 90^\circ\), find \(\tan^{10}\theta + \cot^{10}\theta\). Answer: A. 2 Answer: (A) 2 Let t = tanθ . t + 1/t = 2 gives (t - 1)² = 0 , so t = 1 (θ = 45°). Then tan¹⁰θ + cot¹⁰θ = 1 + 1 = 2 . 1