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

brazil-enem 2024 Q156 Trigonometric Identity Simplification View

In a right triangle, $\sin(\theta) = \dfrac{3}{5}$. What is the value of $\cos(\theta)$?
(A) $\dfrac{1}{5}$
(B) $\dfrac{2}{5}$
(C) $\dfrac{3}{5}$
(D) $\dfrac{4}{5}$
(E) $\dfrac{5}{4}$

cmi-entrance 2016 Q5 4 marks Trigonometric Identity Simplification View

Find the value of the following sum of 100 terms. (Possible hint: also consider the same sum with $\sin^{2}$ instead of $\cos^{2}$.)
$$\cos^{2}\left(\frac{\pi}{101}\right) + \cos^{2}\left(\frac{2\pi}{101}\right) + \cos^{2}\left(\frac{3\pi}{101}\right) + \cdots + \cos^{2}\left(\frac{99\pi}{101}\right) + \cos^{2}\left(\frac{100\pi}{101}\right)$$

csat-suneung 2012 Q27 4 marks Geometric or applied optimisation problem View

As shown in the figure, let Q be the foot of the perpendicular from point P on a circle with center O and diameter AB of length 2 to the line segment AB, let R be the foot of the perpendicular from point Q to the line segment OP, and let S be the foot of the perpendicular from point O to the line segment AP. When $\angle \mathrm { PAQ } = \theta \left( 0 < \theta < \frac { \pi } { 4 } \right)$, let $f ( \theta )$ be the area of triangle AOS and $g ( \theta )$ be the area of triangle PRQ. When $\lim _ { \theta \rightarrow +0 } \frac { \theta ^ { 2 } f ( \theta ) } { g ( \theta ) } = \frac { q } { p }$, find the value of $p ^ { 2 } + q ^ { 2 }$. (Here, $p$ and $q$ are coprime natural numbers.) [4 points]

csat-suneung 2025 Q30C 4 marks Differentiability or Continuity of Trig-Involving Function View

For two constants $a$ ($1 \leq a \leq 2$) and $b$, the function $f(x) = \sin(ax + b + \sin x)$ satisfies the following conditions. (가) $f(0) = 0$ and $f(2\pi) = 2\pi a + b$ (나) The minimum positive value of $t$ such that $f'(0) = f'(t)$ is $4\pi$. Let $A$ be the set of all values of $\alpha$ in the open interval $(0, 4\pi)$ where the function $f(x)$ has a local maximum. If $n$ is the number of elements in set $A$ and $\alpha_{1}$ is the smallest element in set $A$, then $n\alpha_{1} - ab = \frac{q}{p}\pi$. What is the value of $p + q$? [4 points]

gaokao 2010 Q19 Trigonometric Identity Simplification View

19. (Total Score: 12 points) Given $0 < x < \frac { \pi } { 2 }$ , simplify: $\lg \left( \cos x \cdot \tan x + 1 - 2 \sin ^ { 2 } \frac { x } { 2 } \right) + \lg \left[ \sqrt { 2 } \cos \left( x - \frac { \pi } { 4 } \right) \right] - \lg ( 1 + \sin 2 x )$ .

gaokao 2015 Q13 Trigonometric Equation Solving via Identities View

13. Given $\sin \alpha + 2 \cos \alpha = 0$, the value of $2 \sin \alpha \cos \alpha - \cos ^ { 2 } \alpha$ is \_\_\_\_.

gaokao 2015 Q15 Trigonometric Identity Simplification View

15. (This question is worth 13 points) Given the function $f ( x ) = \sqrt { 2 } \sin \frac { x } { 2 } \cos \frac { x } { 2 } - \sqrt { 2 } \sin ^ { 2 } \frac { x } { 2 }$. (I) Find the minimum positive period of $f ( x )$; (II) Find the minimum value of $f ( x )$ on the interval $[ - \pi , 0 ]$.

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 Q19 Multi-Step Composite Problem Using Identities View

19. (This question is worth 12 points)

Let $A , B , C$ be the interior angles of $\triangle A B C$. $\tan A , \tan B$ are the two real roots of the equation $x ^ { 2 } + \sqrt { 3 } p x - p + 1 = 0 ( p \in R )$. (1) Find the size of $C$; (2) If $A B = 3 , A C = \sqrt { 6 }$, find the value of $p$.

gaokao 2015 Q19 Multi-Step Composite Problem Using Identities View

19. As shown in the figure, $A$, $B$, $C$, $D$ are the four interior angles of quadrilateral $ABCD$.
(1) Prove: $\tan \frac { A } { 2 } = \frac { 1 - \cos A } { \sin A }$;
(2) If $A + C = 180 ^ { \circ }$, $AB = 6$, $BC = 3$, $CD = 4$, $AD = 5$, find the value of $\tan \frac { A } { 2 } + \tan \frac { B } { 2 } + \tan \frac { C } { 2 } + \tan \frac { D } { 2 }$. [Figure]

gaokao 2015 Q22 10 marks Triangle Trigonometric Relation View

22. (10 points) Elective 4-1: Geometric Proof As shown in the figure, AB is tangent to circle O at point B. Line AD intersects circle O at points D and E. $\mathrm { BC } \perp \mathrm { DE }$, with C as the foot of the perpendicular. (I) Prove that $\angle \mathrm { CBD } = \angle \mathrm { DBA }$; (II) If $\mathrm { AD } = 3 \mathrm { DC } , ~ \mathrm { BC }

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 2022 Q17 12 marks Determine an angle or side from a trigonometric identity/equation View

Let the sides opposite to angles $A , B , C$ of $\triangle A B C$ be $a , b , c$ respectively. Given
$$\sin C \sin ( A - B ) = \sin B \sin ( C - A )$$
(1) If $A = 2 B$ , find $C$ ;
(2) Prove: $2 a ^ { 2 } = b ^ { 2 } + c ^ { 2 }$ .

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 Q11 6 marks Determine an angle or side from a trigonometric identity/equation View

Given that the area of $\triangle ABC$ is $\frac{1}{4}$, if $\cos 2A + \cos 2B + 2\sin C = 2$, $\cos A \cos B \sin C = \frac{1}{4}$, then
A. $\sin C = \sin^2 A + \sin^2 B$
B. $AB = \sqrt{2}$
C. $\sin A + \sin B = \frac{\sqrt{6}}{2}$
D. $AC^2 + BC^2 = 3$

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 2010 QII.A.1 Direct Proof of an Inequality View

We seek to show that the inequality $|\sin(n\theta)| \leqslant n \sin(\theta)$ is satisfied for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2}\right]$.
a) Show that $\sin(n\theta) \leqslant n \sin(\theta)$ for all $n \in \mathbb{N}^*$ and all $\theta \in \left[0, \frac{\pi}{2n}\right]$.
b) Show that, for all $\theta \in \left[0, \frac{\pi}{2}\right]$, we have $\sin(\theta) \geqslant \frac{2}{\pi} \theta$.
c) Deduce that: $$\forall \theta \in \left[\frac{\pi}{2n}, \frac{\pi}{2}\right], \quad 1 \leqslant n \sin(\theta)$$
d) Conclude.
e) For which values of $\theta \in \left[0, \frac{\pi}{2}\right]$ do we have $|\sin(n\theta)| = n \sin(\theta)$?

grandes-ecoles 2013 QIII.D.6 Evaluation of a Finite or Infinite Sum View

Show that $\prod _ { k = 1 } ^ { n - 1 } \sin \frac { k \pi } { 2 n } = \frac { \sqrt { n } } { 2 ^ { n - 1 } }$.

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 $$f\left(\frac{x}{2}\right) + f\left(\frac{1+x}{2}\right) = 2f(x).$$

iran-konkur 2019 Q125 Triangle Trigonometric Relation View

125. In the figure below, chord $AB$ equals the radius of the circle and $AB \parallel CD$, angle $\beta = 2\alpha$, and $CX$ is tangent to the circle. How many degrees is $\widehat{BD}$?
\begin{minipage}{0.35\textwidth} [Figure: Circle with chord AB equal to radius, AB$\parallel$CD, tangent CX, angles $\alpha$ and $\beta$ marked] \end{minipage} \begin{minipage}{0.55\textwidth} (1) $50$
(2) $60$
(3) $70$
(4) $75$ \end{minipage}

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LFM Pure

Straight Lines & Coordinate Geometry 242

Circles 702

Simultaneous equations 134

Proof 868

Proof by induction 36

Sine and Cosine Rules 185

Radians, Arc Length and Sector Area 22

Trig Graphs & Exact Values 95

Standard trigonometric equations 142

Trig Proofs 92

19. (This question is worth 12 points)

Quadratic trigonometric equations 8

Trigonometric equations in context 11

Modulus function 37

Matrices 1085

Linear transformations 72

Invariant lines and eigenvalues and vectors 339

Reciprocal Trig & Identities 63

Addition & Double Angle Formulae 88

Harmonic Form 14

Small angle approximation 14

Chain Rule 281

Product & Quotient Rules 13

Differentiating Transcendental Functions 160

Connected Rates of Change 48

Standard Integrals and Reverse Chain Rule 82

Integration by Substitution 213

Integration by Parts 44

Implicit equations and differentiation 27

Differential equations 308

3x3 Matrices 164