Trigonometry & Inverse Trigonometry Questions (1013)

The number of solutions of the equation \(\tan x + \sec x = 2\cos x\) lying in the interval \([0, 2\pi]\) is

Ex. 23: Statement I If the sides of a triangle are 13, 14, 15 then the radius of incircle = 4Statement II In triangle ABC, \(A = \sqrt{s(s-a)(s-b)(s-c)}\) where \(s = \dfrac{a+b+c}{2}\) and \(r = \dfrac{A}{s}\)

If in a triangle ABC, \(\cos A \cdot \cos B + \sin A \cdot \sin B \cdot \sin^n C = 1\), \(n \in N\), then prove that the sides are in the ratio \(1:1:\sqrt{2}\).Find the value of \(n\).

A tower stands at the centre of a circular park. A and C are two points on the boundary of the park such that AB subtends an angle of 45° and CB subtends an angle of 30° at the foot of the tower, where B is the foot of the tower. If the radius of the park is 18 m, then the height of the tower (in m) is:

In a triangle \(ABC\), medians \(AD\) and \(BE\) are drawn. If \(AD = 4\), \(\angle DAB = \dfrac{\pi}{6}\) and \(\angle ABE = \dfrac{\pi}{3}\), then the area of the \(\triangle ABC\) is

Let \(S_1\) and \(S_2\) be the areas of inscribed and circumscribed polygons of 10 sides respectively and \(S_3\) is the area of regular polygon of 20 sides inscribed in a circle, then

The product of the sines of the angles of a triangle is \(p\) and the product of their cosines is \(q\). Then, the tangents of the angles are the roots of the equation

The angles A, B and C of a ΔABC are in AP and a : b = 1 : \(\sqrt{3}\). If c = 4 cm, then the area (in sq cm) of this triangle is

If the angles A, B and C of triangle ABC are in arithmetic progression and a, b, c represents length of sides opposite to angles A, B and C respectively, then the value of \(\dfrac{a+c}{\sqrt{(a^2 - ac + c^2)}}\) is:

Let tan−1 y = tan−1 x + tan−1 \left(\frac{2x}{1-x^2}\right), where |x| . Then, a value of y is

The value of \[\frac{\tan A}{1 - \cot A} + \frac{\cot A}{1 - \tan A}\] is equal to

Let \(0 and \(x = X \cos \theta + Y \sin \theta\), \(y = X \sin \theta - Y \cos \theta\) such that \(x^2 + 2xy + y^2 = aX^2 + bY^2\), where \(a\) and \(b\) are constants. Then

The value of \(\displaystyle\sum_{r=0}^{10} \cos^3\dfrac{r\pi}{3}\) is equal to \(\dfrac{-a}{b}\), then the value of \(b\) is (where g.c.d of \((a,b)\) is 1).

In any triangle, if (\(\sin A + \sin B + \sin C\))(\(\sin A + \sin B - \sin C\)) = 3\(\sin A\sin B\),then find the angle \(\frac{C}{10}\) (in degree).

Let \(A_1, A_2, A_3, \ldots, A_n\) be the vertices of an \(n\)-sided regular polygon such that \(\dfrac{1}{A_1 A_2} = \dfrac{1}{A_1 A_3} + \dfrac{1}{A_1 A_4}\). Find the value of \(n\).

If cot (α + β) = 0, then sin (α + 2β) = ?

If a, b, A are given and \(b_1, b_2\) are two values of the third side b such that \(b_2 = 2b_1\). Then, \(\sin A\) is equal to

In triangle ABC, if \(\tan\frac{A}{4} + \tan\frac{B}{4} + \tan\frac{C}{4} = 1\), then triangle ABC is

In triangle ABC, if \(a^2 + c^2 = 2002b^2\), then \(\frac{\cot A + \cot C}{\cot B}\) equals

If \(f_4(x) - f_6(x) = \frac{1}{4}(\sin^4 x + \cos^4 x) - \frac{1}{6}(\cos^6 x + \sin^6 x)\), then the value of this expression equals:

In a right-angled isosceles triangle \(\Delta ADE\) with \(AE = 10\), so that \(AD = DE = 5\sqrt{2}\). In right triangle \(ACD\), \(\tan\beta = \dfrac{CD}{AD}\). Then the area of \(\Delta ABC\) is:

Given \(\sin 3x = \cos 2x\), the number of solutions in \(x \in \left(\dfrac{\pi}{2}, \pi\right)\) is:

If cos θ = 1/(x + 1/(2x)), then what is 2/(x² + 1/x) equal to

If cos 2θ = (√2 + 1) cos θ - 1/√2, then the value of θ is

A person standing on the bank of a river observes that the angle of elevation of the top of a tree on the opposite bank of the river is 60° and when he retires 40 metre away from the tree the angle of elevation becomes 30°. The breadth of the river is:

If a cos³ α + 3a cos α sin² α = m and a sin³ α + 3a cos² α sin α = n, then (m + n)^(2/3) + (m - n)^(2/3) is equal to

If P = cos(cos x) + sin(cos x), then the least and greatest value of P respectively are:

The perimeter of a △ABC is 48 cm and one side is 20 cm. Then remaining sides of △ABC must be greater than:

In triangle ABC, if ∠A = 30°, b = 10 and a = x, then the values of x for which there are 2 possible triangles is given by (All symbols have usual meaning in a triangle)

In triangle ABC, if AC = 8, BC = 7, and D lies between A and B such that AD = 2, BD = 4, then the length CD equals

In a triangle with one angle π/3, the lengths of the sides form an A.P. If the length of the greatest side is 7 cm, the radius of the circumcircle of the triangle is

The value of the expression \[\tan\!\left(\tan^{-1}\!\left(\frac{1}{2}\right)+\tan^{-1}\!\left(\frac{2}{9}\right)+\tan^{-1}\!\left(\frac{1}{8}\right)+\tan^{-1}\!\left(\frac{2}{25}\right)+\tan^{-1}\!\left(\frac{1}{18}\right)+\cdots\cdots\infty\right)\] is:

Determine the smallest positive value of x (in degrees) for which \(\tan(x + 100°) = \tan(x + 50°) \cdot \tan x \cdot \tan(x - 50°)\).

Consider an obtuse angled triangle with sides 8 cm, 15 cm and \(x\) cm (largest side being 15 cm). If \(x\) is an integer, then find the number of possible triangles.

If a, b, c are the sides of a triangle, then the minimum value of \(\dfrac{2a}{b+c-a} + \dfrac{2b}{c+a-b} + \dfrac{2c}{a+b-c}\) is

The general solution-set of the equation \(\cos x + \cos 5x = 2\) is:

Given, $\frac{b+c}{11} = \frac{c+a}{12} = \frac{a+b}{13}$ for a triangle ABC with $\frac{\cos A}{\alpha} = \frac{\cos B}{\beta} = \frac{\cos C}{\gamma}$, then the ordered triad $(\alpha, \beta, \gamma)$ has a value

The equation \((\cos p - 1)x^2 + \cos p \cdot x + \sin p = 0\) where \(x\) is a variable, has real roots. Then the interval of possible values of \(p\) is

A train travelling on one of two intersecting railway lines, subtends at a certain station on the other line, an angle \(\alpha\) when the front of the carriage reaches the junction and an angle \(\beta\) when the end of the carriage reaches it. The two lines are inclined to each other at an angle \(\theta\). Then \(2\cot\theta\) is equal to

Consider a triangle ABC and let a, b and c denote the lengths of the sides opposite to vertices A, B and C, respectively. If a = 1, b = 3 and C = 60°, then \(\sin^2 B\) is equal to

The number of solutions of \(\sin 3x = \cos 2x\), in the interval \(\left(\dfrac{\pi}{2}, \pi\right)\) is

An aeroplane flying at a height of \(\sqrt{3}\) km above the ground passes vertically above another plane at an instant when the angles of elevation of the two planes from a point on the ground are \(60°\) and \(30°\) respectively. The distance (in km) between the two planes at that instant is:

If sin A / sin B = 5/2 and cos A / cos B = 3/2, where 0 , then

If 0 x x which satisfy the equation cos x + cos 2x + cos 3x + cos 4x = 0 is:

If $A(n) = (\sin 1) \times (\sin 2) \times \cdots \times \sin(n), \forall n \in \mathbb{N}$, then the number of elements in the set $A = \{f(1), f(2), \ldots, f(6)\}$ that are positive are

If \(\tan(\alpha - \beta) = \dfrac{\sin(2\beta)}{3 - \cos(2\beta)}\), then \(\tan\alpha = f(\beta)\). The value of \(f\!\left(\dfrac{\pi}{3}\right)\) equals:

If x and y are non-zero real numbers satisfying xy(x2 − y2) = x2 + y2, then find the minimum value of x2 + y2.

The expression cos²(A - B) + cos² B - 2cos(A - B)cos A cos B is

Let x + y + z = θ and k = 2. If \(\cos x + \cos y + \cos z = k\cos\theta\) and \(\sin x + \sin y + \sin z = k\sin\theta\), find the value of \(\cos(x+y) + \cos(y+z) + \cos(z+x)\).

If \(A\), \(B\), \(C\) are in AP and \(B = \frac{\pi}{4}\) then \(\tan A \cdot \tan B \cdot \tan C =\) ______