Trigonometry practice problems

Q1. In \(\triangle ABC\), \(a = 5\), \(b = 3\) and \(c = 7\). Then the value of \[ 3\cos C + 7\cos B \] is equal to:
Answer: 5

Q2. In \(\triangle ABC\), \(a = 1\), \(b = 2\) and \(\angle C = 60^\circ\). If the area of \(\triangle ABC\) is \(\Delta\), then find \[ 2\Delta\sqrt{3} \]
Answer: 3

Q3. In \(\triangle ABC\), if \(b = 20\), \(c = 21\) and \(\sin A = \tfrac{3}{5}\), then \[ a =\;? \]
Answer: 13

Q4. If the sides of a triangle are \(5K\), \(6K\), \(5K\) and the in-radius is \(6\), then the value of \(K\) is:
Answer: 4

Q5. In \(\triangle ABC\), \(a = 6\), \(b = 8\), \(c = 10\). Then the value of \[ \frac{4R}{r} \] is (where all symbols have their usual meaning):
Answer: 10

Q6. In \(\triangle ABC\), \[ a(b^2 + c^2)\cos A \;+\; b(c^2 + a^2)\cos B \;+\; c(a^2 + b^2)\cos C \;=\;\lambda\,abc, \] then the value of \(\lambda\) is:
Answer: 3

Q7. In \(\triangle ABC\), with usual notations, if \(8b = 5c\) and \(\angle C = 2\angle B\), then the value of \[ 25\cos C \] is equal to:
Answer: 7

Q8. The diameter of the circum-circle of a triangle with sides \(61\), \(60\), \(11\) is:
Answer: 61

Q9. In \(\triangle ABC\), \(a = 5\), \(b = 7\) and \(\sin A = \tfrac{3}{4}\). How many such triangles are possible?
Answer: 0

Q10. In \(\triangle ABC\), \(a = 6,\; b = 8,\; c = 10\). Then the area \(\Delta\) of \(\triangle ABC\) is:
Answer: 24

Q11. In \(\triangle ABC\), if \[ a = 8,\; b = 9,\; c = 10, \] then \[ \frac{\tan C}{\sin B} = \frac{p}{q}, \quad p,q\in\mathbb{N}, \] find \(p+q\):
Answer:41

Q13. In \(\triangle ABC\), \[ AB = \sqrt{23},\quad BC = 3,\quad CA = 4. \] Then the value of \[ \frac{\cot A + \cot C}{\cot B} \] is:
Answer: 2

Q14. If in \(\triangle ABC\), \(\angle A = 60^\circ\), then find the value of \[ \bigl(1 + \frac{a}{c} + \frac{b}{c}\bigr) \bigl(1 + \frac{c}{b} - \frac{a}{b}\bigr). \]
Answer: 3

Q15. In \(\triangle ABC\) with usual notations, \(a = 3,\; b = 4,\; c = 5\). Then the in-radius \(r\) is equal to:
Answer: 1

Q16. Let \(a,b,c\) be positive real numbers such that \[ a^2 + ab + b^2 = 25,\quad b^2 + bc + c^2 = 49,\quad c^2 + ca + a^2 = 64. \] Then \[ (a + b + c)^2 =\;? \]

Answer: 129

Q17. In \(\triangle ABC\), \[ (b^2 - c^2)\cot A \;+\;(c^2 - a^2)\cot B \;+\;(a^2 - b^2)\cot C \;=\;? \]

Answer: 0

Q18. Let the sides of a triangle be in the ratio \(3:5:7\) and the largest angle of the triangle is \(\tfrac{k\pi}{12}\). Then the value of \(k\) is:

Answer: 8

Q19. Number of integers in the range of \[ \frac{6}{1 + \tan^2 x} \;+\;\frac{8}{1 + \cot^2 x} \] is:

Answer: 3

Q20. If \(0 \le \theta \le \pi\) and \[ 81\sin^2\theta + 81\cos^2\theta = 30, \] then the number of values of \(\theta\) is:

Answer: 4

Q21. If \(f(x) = 3\sin x - 4\cos x\) and if \(a\) & \(b\) are the maximum and minimum values of \(f(x)\), then \[ \bigl|a - b\bigr| \] equals:

Answer: 10

Q22. The expression \[ \frac{\sin^4 t + \cos^4 t - 1}{\sin^6 t + \cos^6 t - 1} \] when simplified reduces to \(\frac{x}{y}\). Then the value of \(x + y\) is \(\underline{\phantom{abc}}\) (where \(x,y\) are co-prime).

Answer: 5

Q23. The maximum value of the expression \[ \frac{1}{11\sin^2\theta \;+\;24\sin\theta\cos\theta\;+\;29\cos^2\theta} \] is \(\tfrac{1}{t}\). Then the value of \(t\) is \(\underline{\phantom{abc}}\).

Answer: 5

Q24. If \[ (\sin x + \cos x)^2 + k\,\sin x\cos x = 1 \] is true for all \(x\in\mathbb{R}\), then \(|k|\) is \(\underline{\phantom{abc}}\).

Answer: 2

Q25. The value of the expression \[ \sin^6\theta + \cos^6\theta \;+\;3\sin^2\theta\cos^2\theta \] is \(\underline{\phantom{abc}}\).

Answer: 1

Q26. If \(\sin\theta + \cos\theta = 1\), then the value of \(\sin2\theta\) is \(\underline{\phantom{abc}}\).

Answer: 0

Q27. The maximum value of \[ 12\sin\theta \;-\;9\sin^2\theta \] is \(\underline{\phantom{abc}}\).

Answer: 4

Q28. The value of \(\tan75^\circ - \cot75^\circ\) is \(\underline{\phantom{abc}}\).
Answer: \[ 2\sqrt{3}\]

Solution for question 16:









Solution for Question 20:










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