Trigonometry & Inverse Trigonometry Questions (1013)

169. If \(\sin\alpha + \sin\beta + \sin\gamma = -3\), \(\alpha, \beta, \gamma \in (0, 2\pi)\), then \(\cos 2\alpha + \cos 4\beta + \cos 6\gamma\) is equal to:

$AB$ is a vertical tower. The point $A$ is on the ground and $C$ is the middle point of $AB$. The part $CB$ subtends an angle $\alpha$ at a point $P$ on the ground. If $AP = n \cdot AC$, then the correct relation is

If \(\alpha\) is an integer satisfying \(|\alpha| \leq 5 - |[x]|\), where \(x\) is a real number for which \(2x\tan^{-1}x\) is greater than or equal to \(\ln(1+x^2)\), then the number of maximum non-negative possible values of \(\alpha\) is (where \([.]\) denotes the greater integer function)

If \sin 3a = 4\sin a \sin(x+a)\sin(x-a), then x is equal to

99. ABC is a right-angled isosceles triangle in which ∠A = 90°. The midpoint D of AB is joined to C. The ratio of cot ∠DCA and cot ∠DCB is

If \(0 , \(0 and \(\cos x \cdot \sin y = 1\), then find the possible number of values of the ordered pair \((x, y)\).

The angles of triangle PQR are

If \(\tan\theta + \tan 2\theta + \sqrt{3}\tan\theta \tan 2\theta = \sqrt{3}\), then

The set of all values of $\lambda$ for which the equation $\cos^2 2x - 2\sin^4 x - 2\cos^2 x = \lambda$ has a real solution $x$ is

In triangle ABC, \(\angle B (a) \(\frac{\pi}{3}\)(b) \(\frac{\pi}{4}\)(c) \(\frac{\pi}{6}\)(d) \(\frac{\pi}{2}\)

Let maximum value of the expression \(y = |k - 3|\cos 2x + |t - 4|\sin 2x + 3\) where \(0 \leq k \leq 6\) and \(1 \leq t \leq 7\) is equal to 6, then find the minimum value of \((k^2 + t^2)\).

If \(\sin(\alpha+\beta)=1\), \(\sin(\alpha-\beta)=\dfrac{1}{2}\), then \(\tan(\alpha+2\beta)\tan(2\alpha+\beta)\) is equal to

If the equation \(\sum_{n=0}^{10} \operatorname{arc}\cot\left(\frac{1+2^{2n+1}}{2^n}\right) = \operatorname{arc}\cot\frac{a}{b}\), where \(a\) and \(b\) are coprime positive integers. The value of \(\log_2\left(\frac{b+a}{a-b}\right)\), is:

The value of \sqrt{1 + \cos\frac{\pi}{8}} \cdot \sqrt{1 + \cos\frac{3\pi}{8}}\ is equal to

If the angle of elevation of a cloud from a point P which is 25 m above a lake is 30° and the angle of depression of reflection of the cloud in the lake from P be 60°, then the height of the cloud (in meters) from the surface of the lake is __________ .

A triangle has sides 6, 7, 8. The line through its incentre parallel to the shortest side is drawn to meet the other two sides at P and Q. The length of the segment PQ is:

If \alpha + \beta = \frac{\pi}{2}\ and \beta + \gamma = \alpha\, then the value of \tan\alpha\ is

Find the value of \(16(\sin^2 18° + \sin^2 36° + \sin^2 54° + \sin^2 72°)\).

974. Given that \(x \in \mathbb{R}\), find the minimum value of \(\left(3\sqrt{5 - 4\cos x} + \sqrt{13 - 12\sin x}\right)^2\).

If \(M = (\cos^2\theta - 2\cos\theta)\sec^2\phi + 9\text{cosec}^2\phi + 5\sec^2\phi\) where \(\theta \in [0,\ \pi]\) and \(\phi \in \left(0,\ \dfrac{\pi}{2}\right)\), then find the least value of \(M\).

If \(4\sin^4 x + \cos^4 x = 1\), then \(x\) is

The number of real solutions for x from \(2\cos\dfrac{x}{2} = 2^x + 2^{-x}\) is ______.

Question 586. The value of 'm' is equal to:

The number of solutions of the equation \(1 + \sin^4 x - \cos^2 3x\), \(x \in \left[-\dfrac{5\pi}{2}, \dfrac{5\pi}{2}\right]\) is __________ .

If \cos(\theta - \alpha) = a\ and \cos(\theta - \beta) = b\, then \sin^2(\alpha - \beta) + 2ab\cos(\alpha - \beta)\ is equal to

Circumradius of an isosceles △ABC with ∠A = ∠B is 4 times its inradius, then cos A is root of the equation:

Let $\overrightarrow{AB}=-2\hat{i}+\hat{j}+3\hat{k}$, $\overrightarrow{CB}=\alpha\hat{i}+\beta\hat{j}+\gamma\hat{k}$, $\overrightarrow{CA}=4\hat{i}+3\hat{j}+\delta\hat{k}$. If $\delta>0$ and area of $\triangle ABC=5\sqrt{6}$, then $\overrightarrow{CB}\cdot\overrightarrow{CA}$ is equal to

The difference between the greatest and the least possible value of the expression $3 - \cos a + \sin^2 a$ is

If \sec x \cos 5x = -1 and 0 , then x is equal to

If \(\alpha\) satisfies the equation \(2\sqrt{2}\tan^3 x - 54\sqrt{2}\cot^3 x = 19\), then possible value of \((2\tan^2\alpha + \sqrt{2}\tan\alpha)\) can be equal to:

From the top $A$ of a vertical wall $AB$ of height 30 m, the angles of depression of the top $P$ and bottom $Q$ of a vertical tower $PQ$ are $15°$ and $60°$ respectively, $B$ and $Q$ are on the same horizontal level. If $C$ is a point on $AB$ such that $CB=PQ$, then the area (in m²) of the quadrilateral $BCPQ$ is equal to

The maximum value of \(5\sin\theta + 3\sin(\theta - \alpha)\) is 7, then the set of all possible values of \(\alpha\) is

Suppose that the side lengths of a triangle are three consecutive integers and one of the angles is twice another. The number of such triangles is are

In a triangle ABC, (a + b + c)(b + c - a) = kbc, then which of the following is true?

If A + B + C = 180°, then find the value of cos A}{sin B sin C} + cos B}{sin C sin A} + cos C}{sin A sin B}.

In a △ABC if \(9(a^2 + b^2) = 17c^2\) then the value of the expression \(\frac{\cot A + \cot B}{\cot C}\) is:

Let 10 vertical poles standing at equal distances on a straight line, subtend the same angle of elevation \(\alpha\) at a point \(O\) on this line and all the poles are on the same side of \(O\). If the height of the longest pole is '\(h\)' and the distance of the foot of the smallest pole from \(O\) is '\(a\)'; then the distance between two consecutive poles, is

If $A$, $B$, $C$ are in arithmetic progression and $B = \frac{\pi}{4}$, then $A \tan B \tan C =$

If 0 x

Ex. 11: In a triangle ABC, if \(4 \cos A \cos B + 4\sin A \sin B \sin C = 4\), then triangle ABC is

If \(x\), \(y\) and \(z\) are real numbers that satisfy the three equations \[\begin{cases} \tan(x) + \tan(y) + \tan(z) = 6 - (\cot(x) + \cot(y) + \cot(z)) \\ \tan^2(x) + \tan^2(y) + \tan^2(z) = 6 - (\cot^2(x) + \cot^2(y) + \cot^2(z)) \\ \tan^3(x) + \tan^3(y) + \tan^3(z) = 6 - (\cot^3(x) + \cot^3(y) + \cot^3(z)) \end{cases}\] Find the value of the expression \(\left(\dfrac{\tan(x)}{\tan(y)} + \dfrac{\tan(y)}{\tan(z)} + \dfrac{\tan(z)}{\tan(x)} + 3\tan(x)\tan(y)\tan(z)\right)\).

The value of $(1 + \cos \frac{2\pi}{7})(1 + \cos \frac{4\pi}{7})(1 + \cos \frac{6\pi}{7})$ is equal to

\frac{\cos 2x - 3\cos x + 1}{(\cot 2x - \cot x)\sin(x - \pi)} = 0 holds if

If \sin^{100} \theta - \cos^{100} \theta = 1, then \theta is

If for some \alpha, \beta; \alpha \le \beta, \alpha + \beta - 8 and sec (tan 2 -1 2 \alpha) + cosec (cot -1 \beta) - 36 , then \alpha + \beta is_______. 2

If \alpha, \beta, \gamma \in \left(0, \frac{\pi}{2}\right)\, then the value of \frac{\sin(\alpha + \beta + \gamma)}{\sin\alpha + \sin\beta + \sin\gamma}\ is

Given the equation \(8\cos x\left[\cos\left(\dfrac{\pi}{6}+x\right)\cdot\cos\left(\dfrac{\pi}{6}-x\right)-\dfrac{1}{2}\right]=1\), find the sum of all solutions in \([0, \pi]\) and express in the form \(k\pi\). What is \(k\)?

In the interval \(\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]\) the equation \(\log_{\sin}(\cos 2\theta) = 2\) has

The equation \(\sin x - 3\sin 2x + \sin 3x = \cos x - 3\cos 2x + \cos 3x\) has solution

If $\sin x + \sin^2 x = 1$, $x \in \left(0, \dfrac{\pi}{2}\right)$, then $\left(\cos^{12} x + \tan^{12} x\right) + 3\left(\cos^{10} x + \tan^{10} x + \cos^8 x + \tan^8 x\right) + \left(\cos^6 x + \tan^6 x\right)$ is equal to