Trigonometry Questions (1127)

In △ABC, angle A is 120°, BC + CA = 20 and AB + BC = 21, then the length of the side BC equals:

Let $x_1$ and $x_2$ ($x_1 > x_2$) are the roots of the equation $9^{\log_9(x^2-4x+5)} = x-1$, then the value of $\tan(x_1)\pi + \sec(x_2)\pi$ is:

The solution of the equation \(k \cos x - 3 \sin x = k + 1\) is possible only if(JEE Main 2019)

If \(4 \sin 27° = \sqrt{a + \sqrt{b}}\), then the value of \((a + b - ab + 2)^4\) must be:

Ex. 35. Statement I: Common value(s) of x satisfying the equations logsinx(secx+8)>0 and logsinxcosx+logcosxsinx=2 in (0,4π) does not exist.Statement II: On solving above trigonometric equations we have to take intersection of trigonometric chains given by secx>1 and x=nπ+π4, n∈I.

If H is the orthocentre of triangle ABC, R = circumradius and P = AH + BH + CH, then

The number of ordered pairs (x, y) satisfying |x| + |y| = 2 and sin\left(\frac{\pi x^2}{4}\right) = 1.

Given a = 6, b = 3 and \(\cos(A - B) = -\frac{1}{4}\), find the area of the triangle.

Question 585. The value of M is:

If in a triangle ABC, sin A + sin B + sin C}{sin A + sin B - sin C} = 2X cot A}{2} cot B}{2}, then find the value of X.

$\tan^{-1}\left(\tan\dfrac{5\pi}{6}\right)+\cos^{-1}\left(\cos\dfrac{13\pi}{6}\right)=$

If $r_1$ and $r_2$ are the remainder when $f(x)=4x^3+3x^2-12x+a$ is divided by $(x-1)$ and $(x+2)$, and $2r_1+r_2=6$, then $a=$

If \sin^{-1}\frac{\alpha}{17} + \cos^{-1}\frac{1}{5} - \tan^{-1}\frac{\alpha}{36} = 0, \quad 0 < \alpha < 13, \text{ then } \sin^{-1}(\sin\alpha) + \cos^{-1}(\cos\alpha) \text{ is equal to}

11. Consider the function \(f(x) = \frac{\sqrt{1 + \cos x} + \sqrt{1 - \cos x}}{\sqrt{1 + \cos x} - \sqrt{1 - \cos x}}\). If \(x \in (\pi, 2\pi)\), then \(f(x)\) is:

Let \(\sec x + \tan x = \frac{22}{7}\), where \(0

In an acute angled triangle ABC, given that a = 6, b = 3 and \(\cos(A - B) = -\frac{1}{4}\), find angle C.

\(2\sin^{-1}\sqrt{\dfrac{1-x}{2}} = \cos^{-1}(\underline{\quad})\).

For \(f(x)=e^x\), \(g(x)=\sin^{-1}x\), which are necessarily true?

Find the radius of the circle escribed to the triangle ABC on the side BC if $\angle NAB = 30°$; $\angle BAC = 30°$; $AB = AC = 5$.

In a $\triangle ABC$, $\angle A > \angle B$. Let $\angle A, \angle B$ satisfy the equation $3\sin x - 4\sin^3 x - k = 0$, where $0 < k < 1$, then $\angle C$ is equal to:

An aeroplane flying horizontally 1 km above the ground is observed at an elevation of 60° and after 10 seconds the elevation is observed to be 30°. The uniform speed of the aeroplane in km/h is

The least positive value of $x$ satisfying the equation $\dfrac{\sin x}{\cos 3x} + \dfrac{\sin 3x}{\cos 9x} + \dfrac{\sin 9x}{\cos 27x} = 0$ is:

If sin x + cos x = 1 + sin x cos x, then x is

Let ABC be a triangle such that ∠A = 45°, ∠B = 75°, then \(a + c\sqrt{2}\) is equal to

Let $x=\sin1°$. The value of $\dfrac{1}{\cos0°\cos1°}+\dfrac{1}{\cos1°\cos2°}+\cdots+\dfrac{1}{\cos44°\cos45°}$ is

The maximum value of the function $f(x) = \dfrac{4\cot^{-1}x}{\pi} - \dfrac{\pi}{4\cot^{-1}(-x)}$ occurs at $x$ equal to

If $(\sin^{-1}x)^2 + (\sin^{-1}y)^2 + 2\sin^{-1}x\sin^{-1}y = \pi^2$, then $x^2 + y^2$ is equal to:

If \(\cos(x-y)\), \(\cos x\) and \(\cos(x+y)\) are in HP then \(\cos x \cdot \sec\left(\frac{y}{2}\right) =\) ______

If $\dfrac{\cos x + \cos y + \cos z}{\cos(x+y+z)} = 2$ and $\dfrac{\sin x + \sin y + \sin z}{\sin(x+y+z)} = 2$, then the value of $\cos(x+y) + \cos(y+z) + \cos(z+x)$ is equal to: (where $x, y, z \in R$)

The value of $4\cos\dfrac{\pi}{10} - 3\sec\dfrac{\pi}{10} - 2\tan\dfrac{\pi}{10}$ is equal to

If $x = \sin^{-1}(\sin 10)$ and $y = \cos^{-1}(\cos 10)$, then $y - x$ is equal to:

In a triangle \(ABC\), it is given that \(\dfrac{[ABC]}{R} = 4\), where \(R\) is the circumradius. Show that \(\sum_{\text{cyc}} a\cos A = 4R \prod_{\text{cyc}} \sin A\) and find \(\prod_{\text{cyc}} \sin A\).

In $\triangle ABC$, $\angle A = \tan^{-1}7$, $\angle C = \tan^{-1}\frac{4}{3}$. Let $D$ be an interior point on side $AC$ such that area of $\triangle ABD$ is twice the area of $\triangle BCD$. If $\angle ABD = \theta$, then the value of $\tan 2\theta$ is

The value of $\displaystyle\sum_{r=2}^{\infty} \cot^{-1}(r^2 - 5r + 7)$ is

Ex. 62: If in triangle ABC, $\tan A + \tan B + \tan C = 6$ and $\tan A \tan B = 2$, then $\sin^2 A : \sin^2 B : \sin^2 C$ is

The period of the function \( f(x) = \sin^4 x + \cos^4 x \) is:

If $a=2$, then the sum of the infinite series $\cot^{-1}(2a^{-1}+a)+\cot^{-1}(2a^{-1}+3a)+\cot^{-1}(2a^{-1}+6a)+\cot^{-1}(2a^{-1}+10a)+\cdots$ is

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:

176. If \(\alpha = \sin\theta\,|\sin\theta|\) and \(\beta = \cos\theta\,|\cos\theta|\) where \(\theta \in \left[\dfrac{199\pi}{2},\, 100\pi\right]\), then:

The angle of elevation of tower from a point A due south of it is 30° and from a point B due west of it is 45°. If the height of the tower be 100 m, then AB =

Let \(f(x) = x^4 - 8x^3 + 18x^2 - 6x + 1 - 2\sqrt{3}\), then \(f\left(x = \cot\dfrac{\pi}{12}\right)\) is equal to:

If sin x₁ + sin x₂ + sin x₃ + ... + sin x₂₀₀₈ = 2008, then find the value of sin²⁰⁰⁸ x₁ + sin²⁰⁰⁸ x₂ + sin²⁰⁰⁸ x₃ + ... + sin²⁰⁰⁸ x₂₀₀₈.

In a triangle ABC if \(3 \sin A + 4 \cos B = 6\); \(4 \sin B + 3 \cos A = 1\) then possible value(s) of ∠C be:

The value of the expression $\dfrac{\sin 20°(4\cos 20°+1)}{\cos 20°\cdot\cos 30°}$ is

The maximum value of the function $f(x) = \dfrac{4\cot^{-1}x}{\pi} - \dfrac{\pi}{4\cot^{-1}(-x)}$ occurs at $x$ equal to

If √2 cos A = cos B + cos³ B, and √2 sin A = sin B - sin³ B then sin(A - B) = ?

If 2 tan-1(1/5) - sin-1(3/5) = -cos-1(63/l), then l =

If $y = \tan^{-1}\dfrac{4x}{1+5x^2} + \tan^{-1}\dfrac{2+3x}{3-2x}$, find $\dfrac{dy}{dx} = \dfrac{\alpha}{1+25x^2}$. Find $\alpha$.

Value of $\sin32°\cdot\sin88°\cdot\sin152°$ equals

If $x=\cos1°\cos2°\cos3°\cdots\cos89°$ and $y=\cos2°\cos6°\cos10°\cdots\cos86°$, then $\dfrac{2}{7}\log_2\!\left(\dfrac{y}{x}\right)$ is equal to