LFM Pure

brazil-enem 2015 Q180 Trigonometric Identity Simplification View

QUESTION 180
The value of $\cos^2 30^\circ + \sin^2 30^\circ$ is
(A) $\frac{1}{2}$
(B) $\frac{\sqrt{3}}{2}$
(C) 1
(D) $\sqrt{3}$
(E) 2

gaokao 2015 Q16 Trigonometric Identity Simplification View

16. Given the function $f ( x ) = ( \sin x + \cos x ) ^ { 2 } + \cos 2 x$
(1) Find the minimum positive period of $f ( x )$;
(2) Find the maximum and minimum values of $f ( x )$ on the interval $\left[ 0 , \frac { \pi } { 2 } \right]$.

gaokao 2015 Q17 Triangle Trigonometric Relation View

17. (This question is worth 12 points) Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively, with $a = b \tan A$. (I) Prove that: $\sin \mathrm { B } = \cos \mathrm { A }$ (II) If $\sin C - \sin A \cos B = \frac { 3 } { 4 }$ and $B$ is an obtuse angle, find $A$, $B$, and $C$.

gaokao 2015 Q17 Triangle Trigonometric Relation View

17. (This question is worth 12 points) Let the sides opposite to angles $A, B, C$ of $\triangle ABC$ be $a, b, c$ respectively. Given $a = b \tan A$ and $B$ is an obtuse angle. (I) Prove: $\mathrm { B } - \mathrm { A } = \frac { \pi } { 2 }$ (II) Find the range of $\sin \mathrm { A } + \sin \mathrm { C }$.

gaokao 2015 Q17 12 marks Triangle Trigonometric Relation View

17. (12 points) In $\triangle \mathrm { ABC }$, the sides opposite to angles $\mathrm { A } , \mathrm { B }$, C are $a , b , c$ respectively. The vector $\vec { m } = ( a , \sqrt { 3 } b )$ is parallel to $\vec { n } = ( \cos \mathrm { A } , \sin \mathrm { B } )$. (I) Find A; (II) If $a = \sqrt { 7 } , b = 2$, find the area of $\triangle \mathrm { ABC }$.

gaokao 2019 Q17 12 marks Triangle Trigonometric Relation View

17. (12 points) In $\triangle A B C$, the angles $A , B , C$ have opposite sides $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$.
(1) Find $A$;
(2) If $\sqrt { 2

gaokao 2019 Q17 12 marks Triangle Trigonometric Relation View

17. (12 points) In $\triangle A B C$ , let the sides opposite to angles $A , B , C$ be $a , b , c$ respectively. Given $( \sin B - \sin C ) ^ { 2 } = \sin ^ { 2 } A - \sin B \sin C$ .
(1) Find $A$ ;
(2) If $\sqrt { 2 } a + b = 2 c$ , find $\sin C$ .

gaokao 2020 Q5 5 marks Trigonometric Identity Simplification View

Given $\sin \theta + \sin \left( \theta + \frac { \pi } { 3 } \right) = 1$, then $\sin \left( \theta + \frac { \pi } { 6 } \right) =$
A. $\frac { 1 } { 2 }$
B. $\frac { \sqrt { 3 } } { 3 }$
C. $\frac { 2 } { 3 }$
D. $\frac { \sqrt { 2 } } { 2 }$

gaokao 2020 Q9 5 marks Trigonometric Equation Constraint Deduction View

Given $\alpha \in ( 0 , \pi )$ and $3 \cos 2 \alpha - 8 \cos \alpha = 5$, then $\sin \alpha =$
A. $\frac { \sqrt { 5 } } { 3 }$
B. $\frac { 2 } { 3 }$
C. $\frac { 1 } { 3 }$
D. $\frac { \sqrt { 5 } } { 9 }$

gaokao 2022 Q18 12 marks Triangle Trigonometric Relation View

18. (12 points) Let the sides opposite to angles $A$, $B$, $C$ of $\triangle A B C$ be $a$, $b$, $c$ respectively. Given that $\frac { \cos A } { 1 + \sin A } = \frac { \sin 2 B } { 1 + \cos 2 B }$.
(1) If $C = \frac { 2 \pi } { 3 }$, find $B$;
(2) Find the minimum value of $\frac { a ^ { 2 } + b ^ { 2 } } { c ^ { 2 } }$ .

gaokao 2023 Q7 5 marks Trigonometric Equation Constraint Deduction View

``$\sin^{2} \alpha + \sin^{2} \beta = 1$'' is ``$\cos \alpha + \cos \beta = 0$'' a
A. sufficient but not necessary condition
B. necessary but not sufficient condition
C. necessary and sufficient condition
D. neither sufficient nor necessary condition

gaokao 2025 Q19 17 marks Extremal Value of Trigonometric Expression View

Let the function $f(x) = 5\cos x - \cos 5x$.
(1) Find the maximum value of $f(x)$ on $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a$ is a real number, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) If there exists $\varphi$ such that for all $x$, $5\cos x - \cos(5x + \varphi) \leq b$, find the minimum value of $b$.

gaokao 2025 Q19 17 marks Extremal Value of Trigonometric Expression View

(17 points)
(1) Find the maximum value of the function $f(x) = 5\cos x - \cos 5x$ on the interval $\left[0, \frac{\pi}{4}\right]$.
(2) Given $\theta \in (0, \pi)$ and $a \in \mathbf{R}$, prove that there exists $y \in [a - \theta, a + \theta]$ such that $\cos y \leq \cos \theta$.
(3) Let $b \in \mathbf{R}$. If there exists $\varphi \in \mathbf{R}$ such that $5\cos x - \cos(5x + \varphi) \leq b$ holds for all $x \in \mathbf{R}$, find the minimum value of $b$.

grandes-ecoles 2019 Q15 Trigonometric Inequality Proof View

Let $\theta \in [-\pi, \pi]$.
a. Show that $\cos(\theta) \geq 1 - \frac{\theta^2}{2}$.
b. Show that $\left|\frac{e^{i\theta} - (1-p)}{p}\right| \leq \exp\left(\frac{1-p}{2p^2} \cdot \theta^2\right)$.
Hint. One may calculate $\left|\frac{e^{i\theta} - (1-p)}{p}\right|^2$.

grandes-ecoles 2019 Q27 Trigonometric Inequality Proof View

Show the inequality $t\cos(t) \leqslant \sin(t)$, for every $t$ in $[0, \pi/2]$.

grandes-ecoles 2022 Q2a Trigonometric Identity Simplification View

Let the functions $f$, $g$ and $D$ be defined on $\mathbb{R} \backslash \mathbb{Z}$ by: $$f(x) = \pi \operatorname{cotan}(\pi x) = \pi \frac{\cos(\pi x)}{\sin(\pi x)}, \quad g(x) = \frac{1}{x} + \sum_{n=1}^{+\infty} \left(\frac{1}{x+n} + \frac{1}{x-n}\right).$$ We set $D = f - g$.
Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$

grandes-ecoles 2022 Q2a Trigonometric Identity Simplification View

Let $f(x) = \pi \operatorname{cotan}(\pi x)$. Show that for all $x \in \mathbb{R} \backslash \mathbb{Z}$, we have $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x)$$

grandes-ecoles 2022 Q8 Norm or Modulus Computation Involving Trig/Complex Exponentials View

Let $x \in [ 0,1 [$ and $\theta \in \mathbf { R }$. Using the function $L$, show that
$$\left| \frac { 1 - x } { 1 - x e ^ { i \theta } } \right| \leq \exp ( - ( 1 - \cos \theta ) x )$$
Deduce that for all $x \in [ 0,1 [$ and all real $\theta$,
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 1 - x } + \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \right)$$

grandes-ecoles 2022 Q9 Norm or Modulus Computation Involving Trig/Complex Exponentials View

Let $x \in \left[ \frac { 1 } { 2 } , 1 [ \right.$ and $\theta \in \mathbf { R }$. Show that
$$\frac { 1 } { 1 - x } - \operatorname { Re } \left( \frac { 1 } { 1 - x e ^ { i \theta } } \right) \geq \frac { x ( 1 - \cos \theta ) } { ( 1 - x ) \left( ( 1 - x ) ^ { 2 } + 2 x ( 1 - \cos \theta ) \right) }$$
Deduce that
$$\left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 - \cos \theta } { 6 ( 1 - x ) ^ { 3 } } \right) \quad \text { or } \quad \left| \frac { P \left( x e ^ { i \theta } \right) } { P ( x ) } \right| \leq \exp \left( - \frac { 1 } { 3 ( 1 - x ) } \right)$$

isi-entrance 2005 Q4 Trigonometric Inequality Proof View

Show that $\sin^5 x + \cos^3 x \geq \sin^3 x + \cos^2 x$ implies the expression equals $1$, and find when equality holds.

isi-entrance 2010 Q6 Extremal Value of Trigonometric Expression View

Let $\alpha$, $\beta$ and $\gamma$ be the angles of an acute angled triangle. Then the quantity $\tan\alpha\tan\beta\tan\gamma$
(a) Can have any real value
(b) Is $\leq 3\sqrt{3}$
(c) Is $\geq 3\sqrt{3}$
(d) None of the above.

isi-entrance 2013 Q49 4 marks Ordering or Comparing Trigonometric Expressions View

Suppose $x, y \in (0, \pi/2)$ and $x \neq y$. Which of the following statement is true?
(A) $2\sin(x + y) < \sin 2x + \sin 2y$ for all $x, y$.
(B) $2\sin(x + y) > \sin 2x + \sin 2y$ for all $x, y$.
(C) There exist $x, y$ such that $2\sin(x + y) = \sin 2x + \sin 2y$.
(D) None of the above.

isi-entrance 2021 Q27 Extremal Value of Trigonometric Expression View

If the maximum and minimum values of $\sin ^ { 6 } x + \cos ^ { 6 } x$, as $x$ takes all real values, are $a$ and $b$, respectively, then $a - b$ equals
(A) $\frac { 1 } { 2 }$.
(B) $\frac { 2 } { 3 }$.
(C) $\frac { 3 } { 4 }$.
(D) 1 .

isi-entrance 2026 Q2 10 marks Trigonometric Equation Constraint Deduction View

If the interior angles of a triangle $ABC$ satisfy the equality, $$\sin ^ { 2 } A + \sin ^ { 2 } B + \sin ^ { 2 } C = 2 \left( \cos ^ { 2 } A + \cos ^ { 2 } B + \cos ^ { 2 } C \right) ,$$ prove that the triangle must have a right angle.

isi-entrance 2026 Q16 Ordering or Comparing Trigonometric Expressions View

Suppose $x , y \in ( 0 , \pi / 2 )$ and $x \neq y$. Which of the following statements is true?
(a) $2 \sin ( x + y ) < \sin 2 x + \sin 2 y$ for all $x , y$.
(b) $2 \sin ( x + y ) > \sin 2 x + \sin 2 y$ for all $x , y$.
(c) There exist $x , y$ such that $2 \sin ( x + y ) = \sin 2 x + \sin 2 y$.
(d) None of the above.

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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