LFM Pure

jee-main 2022 Q72 Inverse trigonometric equation View

The value of $\cot \left( \sum _ { n = 1 } ^ { 50 } \tan ^ { - 1 } \left( \frac { 1 } { 1 + n + n ^ { 2 } } \right) \right)$ is
(1) $\frac { 25 } { 26 }$
(2) $\frac { 50 } { 51 }$
(3) $\frac { 26 } { 25 }$
(4) $\frac { 52 } { 51 }$

jee-main 2022 Q86 Solve trigonometric equation for solutions in an interval View

Let $S = \left[ - \pi , \frac { \pi } { 2 } \right) - \left\{ - \frac { \pi } { 2 } , - \frac { \pi } { 4 } , - \frac { 3 \pi } { 4 } , \frac { \pi } { 4 } \right\}$. Then the number of elements in the set $A = \{ \theta \in S : \tan \theta ( 1 + \sqrt { 5 } \tan ( 2 \theta ) ) = \sqrt { 5 } - \tan ( 2 \theta ) \}$ is $\_\_\_\_$.

jee-main 2022 Q87 Inverse trigonometric equation View

For $k \in \mathbb { R }$, let the solutions of the equation $\cos \left( \sin ^ { - 1 } \left( x \cot \left( \tan ^ { - 1 } \left( \cos \left( \sin ^ { - 1 } x \right) \right) \right) \right) \right) = k , 0 < | x | < \frac { 1 } { \sqrt { 2 } }$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x ^ { 2 } - b x - 5 = 0$ are $\frac { 1 } { \alpha ^ { 2 } } + \frac { 1 } { \beta ^ { 2 } }$ and $\frac { \alpha } { \beta }$, then $\frac { b } { k ^ { 2 } }$ is equal to $\_\_\_\_$ .

jee-main 2023 Q69 Solve trigonometric equation for solutions in an interval View

If $m$ and $n$ respectively are the numbers of positive and negative value of $\theta$ in the interval $[ - \pi , \pi ]$ that satisfy the equation $\cos 2 \theta \cos \frac { \theta } { 2 } = \cos 3 \theta \cos \frac { 9 \theta } { 2 }$, then $m n$ is equal to $\_\_\_\_$.

jee-main 2023 Q70 Inverse trigonometric equation View

If $\sin^{-1}\frac{\alpha}{17} + \cos^{-1}\frac{4}{5} - \tan^{-1}\frac{77}{36} = 0$, $0 < \alpha < 13$, then $\sin^{-1}(\sin\alpha) + \cos^{-1}(\cos\alpha)$ is equal to
(1) $\pi$
(2) 16
(3) 0
(4) $16 - 5\pi$

jee-main 2023 Q71 Inverse trigonometric equation View

Let $y = f(x)$ represent a parabola with focus $\left(-\frac{1}{2}, 0\right)$ and directrix $y = -\frac{1}{2}$. Then $S = \left\{x \in \mathbb{R} : \tan^{-1}\sqrt{f(x)} + \sin^{-1}\sqrt{f(x)+1} = \frac{\pi}{2}\right\}$:
(1) contains exactly two elements
(2) contains exactly one element
(3) is an infinite set
(4) is an empty set

jee-main 2023 Q76 Solve trigonometric equation for solutions in an interval View

Let $S = \{\theta \in [0, 2\pi): \tan(\pi\cos\theta) + \tan(\pi\sin\theta) = 0\}$. Then $\sum_{\theta \in S} \sin\left(\theta + \frac{\pi}{4}\right)$ is equal to $\_\_\_\_$.

jee-main 2023 Q76 Inverse trigonometric equation View

If the sum of all the solutions of $\tan ^ { - 1 } \left( \frac { 2 x } { 1 - x ^ { 2 } } \right) + \cot ^ { - 1 } \left( \frac { 1 - x ^ { 2 } } { 2 x } \right) = \frac { \pi } { 3 } , - 1 < x < 1 , x \neq 0$, is $\alpha - \frac { 4 } { \sqrt { 3 } }$, then $\alpha$ is equal to $\_\_\_\_$ .

jee-main 2024 Q64 Solve trigonometric equation for solutions in an interval View

If $\alpha , - \frac { \pi } { 2 } < \alpha < \frac { \pi } { 2 }$ is the solution of $4 \cos \theta + 5 \sin \theta = 1$, then the value of $\tan \alpha$ is
(1) $\frac { 10 - \sqrt { 10 } } { 6 }$
(2) $\frac { 10 - \sqrt { 10 } } { 12 }$
(3) $\frac { \sqrt { 10 } - 10 } { 12 }$
(4) $\frac { \sqrt { 10 } - 10 } { 6 }$

jee-main 2024 Q64 Solve trigonometric equation for solutions in an interval View

If $2 \tan ^ { 2 } \theta - 5 \sec \theta = 1$ has exactly 7 solutions in the interval $0 , \frac { \mathrm { n } \pi } { 2 }$, for the least value of $\mathrm { n } \in \mathrm { N }$ then $\sum _ { \mathrm { k } = 1 } ^ { \mathrm { n } } \frac { \mathrm { k } } { 2 ^ { \mathrm { k } } }$ is equal to :
(1) $\frac { 1 } { 2 ^ { 15 } } 2 ^ { 14 } - 14$
(2) $\frac { 1 } { 2 _ { 1 } ^ { 14 } } 2 ^ { 15 } - 15$
(3) $1 - \frac { 15 } { 2 ^ { 13 } }$
(4) $\frac { 1 } { 2 ^ { 13 } } 2 ^ { 14 } - 15$

jee-main 2024 Q70 Inverse trigonometric equation View

Considering only the principal values of inverse trigonometric functions, the number of positive real values of $x$ satisfying $\tan ^ { - 1 } ( \mathrm { x } ) + \tan ^ { - 1 } ( 2 \mathrm { x } ) = \frac { \pi } { 4 }$ is :
(1) More than 2
(2) 1
(3) 2
(4) 0

jee-main 2024 Q72 Inverse trigonometric equation View

If $a = \sin^{-1}(\sin 5)$ and $b = \cos^{-1}(\cos 5)$, then $a^2 + b^2$ is equal to
(1) $4\pi^2 + 25$
(2) $8\pi^2 - 40\pi + 50$
(3) $4\pi^2 - 20\pi + 50$
(4) 25

jee-main 2024 Q83 Solve trigonometric equation for solutions in an interval View

Let the set of all $a \in R$ such that the equation $\cos 2 x + a \sin x = 2 a - 7$ has a solution be $[ p , q ]$ and $r = \tan 9 ^ { \circ } - \tan 27 ^ { \circ } - \frac { 1 } { \cot 63 ^ { \circ } } + \tan 81 ^ { \circ }$, then $p q r$ is equal to $\_\_\_\_$.

jee-main 2024 Q83 Solve trigonometric equation for solutions in an interval View

The number of solutions of $\sin ^ { 2 } x + \left( 2 + 2 x - x ^ { 2 } \right) \sin x - 3 ( x - 1 ) ^ { 2 } = 0$, where $- \pi \leq x \leq \pi$, is $\_\_\_\_$

jee-main 2024 Q86 Inverse trigonometric equation View

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $2 \sin ^ { - 1 } x + 3 \cos ^ { - 1 } x = \frac { 2 \pi } { 5 }$, is $\_\_\_\_$

jee-main 2025 Q3 Trigonometric equation with algebraic or logarithmic coupling View

Let $\mathrm{A} = \left\{ x \in (0, \pi) - \left\{ \frac{\pi}{2} \right\} : \log_{(2/\pi)} |\sin x| + \log_{(2/\pi)} |\cos x| = 2 \right\}$ and $\mathrm{B} = \{ x \geqslant 0 : \sqrt{x}(\sqrt{x} - 4) - 3|\sqrt{x} - 2| + 6 = 0 \}$. Then $\mathrm{n}(\mathrm{A} \cup \mathrm{B})$ is equal to:
(1) 4
(2) 8
(3) 6
(4) 2

jee-main 2025 Q18 Solve trigonometric equation for solutions in an interval View

The sum of all values of $\theta \in [ 0,2 \pi ]$ satisfying $2 \sin ^ { 2 } \theta = \cos 2 \theta$ and $2 \cos ^ { 2 } \theta = 3 \sin \theta$ is
(1) $4 \pi$
(2) $\frac { 5 \pi } { 6 }$
(3) $\pi$
(4) $\frac { \pi } { 2 }$

jee-main 2025 Q22 Inverse trigonometric equation View

If for some $\alpha, \beta$; $\alpha \leq \beta$, $\alpha + \beta = 8$ and $\sec^2(\tan^{-1}\alpha) + \operatorname{cosec}^2(\cot^{-1}\beta) = 36$, then $\alpha^2 + \beta$ is $\underline{\hspace{2cm}}$.

kyotsu-test 2010 QCourse2-IV-Q2 Geometric problem using trigonometric function graphs View

Q2 Suppose that the curve $y=2\cos 2x$ and the curve $y=4\cos x+k$ have a common tangent at $x=a\left(0(1) We set $f(x)=2\cos 2x$ and $g(x)=4\cos x+k$. Since we have assumed that $y=f(x)$ and $y=g(x)$ have a common tangent at $x=a$, we see that
$$f'(a)=g'(a), \quad f(a)=g(a).$$
Since $f'(a)=g'(a)$ and $0Hence the coordinates of the tangent point are $\left(\frac{\pi}{\mathbf{O}},-\mathbf{Q}\right)$, and the equation of the common tangent line is
$$y=-\mathbf{R}\sqrt{\mathbf{S}}\left(x-\frac{\pi}{\mathbf{T}}\right)-\mathbf{U}.$$
(2) We are to find the area $S$ of the region bounded by these two curves over the range $-\frac{\pi}{2}\leqq x\leqq\frac{\pi}{2}$.
Since both of these curves are symmetric with respect to the $y$-axis, by putting $b=\mathbf{V}$ and $c=\frac{\pi}{\mathbf{O}}$ we have
$$S=\mathbf{W}\int_{b}^{c}(2\cos 2x-4\cos x-k)\,dx$$
By calculating this, we obtain
$$S=\mathbf{X}\cdot\pi-\mathbf{Y}\cdot\mathbf{Z}.$$

taiwan-gsat 2023 Q18 3 marks Evaluate trigonometric expression given a constraint View

On the coordinate plane, $O$ is the origin, and points $A(1,0)$ and $B(-2,0)$ are given. There are also two points $P$ and $Q$ in the upper half-plane satisfying $\overline{AP} = \overline{OA}$, $\overline{BQ} = \overline{OB}$, $\angle POQ$ is a right angle. Let $\angle AOP = \theta$.
The length of line segment $\overline{OP}$ is which of the following options? (Single choice question, 3 points)
(1) $\sin\theta$
(2) $\cos\theta$
(3) $2\sin\theta$
(4) $2\cos\theta$
(5) $\cos 2\theta$

taiwan-gsat 2024 Q4 5 marks Solve trigonometric equation for solutions in an interval View

How many real numbers $x$ satisfy $\sin\left(x + \frac{\pi}{6}\right) = \sin x + \sin\frac{\pi}{6}$ and $0 \leq x < 2\pi$?
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5 or more

taiwan-gsat 2025 Q1 6 marks Solve trigonometric equation for solutions in an interval View

On the coordinate plane, the graph of the function $y = \sin x$ is symmetric about $x = \frac{\pi}{2}$, as shown in the figure. Find the value of $\theta$ in the range $0 < \theta \leq \pi$ that satisfies $\sin \theta = \sin\left(\theta + \frac{\pi}{5}\right)$.
(1) $\frac{\pi}{5}$
(2) $\frac{2\pi}{5}$
(3) $\frac{3\pi}{5}$
(4) $\frac{4\pi}{5}$
(5) $\pi$

taiwan-gsat 2025 Q20 6 marks Applied trigonometric modeling View

Continuing from question 19, where $f ( x ) = a \sin ( b x )$ with $f(0) = f(12) = 0$ and $f(2) = 4$, a person wants to sunbathe when the UVI value is between $4 \sqrt { 2 }$ and $4 \sqrt { 3 }$ (inclusive). The time during which he can sunbathe is set as $t$ hours after sunrise. Find the maximum possible range of $t$.

turkey-yks 2013 Q23 Solve trigonometric equation for solutions in an interval View

Given that $0 < \mathrm { x } < \pi$,
$$\sin ^ { 4 } x = \cos ^ { 4 } x$$
What is the sum of the $\mathbf { x }$ values that satisfy this equality?
A) $\frac { 3 \pi } { 2 }$
B) $\frac { 4 \pi } { 3 }$
C) $\frac { 5 \pi } { 4 }$
D) $\pi$
E) $2 \pi$

turkey-yks 2014 Q24 Evaluate trigonometric expression given a constraint View

ABCD is a square $| \mathrm { AB } | = 3$ units $| \mathrm { BE } | = | \mathrm { CF } | = 1$ unit $m ( \widehat { F A E } ) = x$
According to the given information above, what is the value of $\cot \mathrm { x }$?
A) $\frac { 6 } { 5 }$
B) $\frac { 8 } { 5 }$
C) $\frac { 7 } { 6 }$
D) $\frac { 9 } { 7 }$
E) $\frac { 11 } { 8 }$

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LFM Pure

Straight Lines & Coordinate Geometry 247

Circles 443

Simultaneous equations 79

Proof 711

Proof by induction 25

Sine and Cosine Rules 195

Radians, Arc Length and Sector Area 35

Trig Graphs & Exact Values 75

Standard trigonometric equations 106

Trig Proofs 53

Quadratic trigonometric equations 36

Trigonometric equations in context 18

Modulus function 49

Matrices 1553

Linear transformations 26

Invariant lines and eigenvalues and vectors 188

Reciprocal Trig & Identities 44

Addition & Double Angle Formulae 98

Harmonic Form 12

Small angle approximation 20

Chain Rule 113

Product & Quotient Rules 18

Differentiating Transcendental Functions 183

Connected Rates of Change 56

Standard Integrals and Reverse Chain Rule 43

Integration by Substitution 116

Integration by Parts 74

Implicit equations and differentiation 43

Differential equations 352

3x3 Matrices 144