LFM Pure

taiwan-gsat 2020 Q1 6 marks Ordering or Comparing Trigonometric Expressions View

Given $45^{\circ} < \theta < 50^{\circ}$, and let $a = 1 - \cos^{2}\theta$, $b = \frac{1}{\cos\theta} - \cos\theta$, $c = \frac{\tan\theta}{\tan^{2}\theta + 1}$. Regarding the relative sizes of the three values $a$, $b$, $c$, select the correct option.
(1) $a < b < c$
(2) $a < c < b$
(3) $b < a < c$
(4) $b < c < a$
(5) $c < a < b$

tmua 2020 Q2 1 marks View

Given that $\tan \theta = 2$ and $180 ^ { \circ } < \theta < 360 ^ { \circ }$, find the value of $\cos \theta$
A $\sqrt { 3 }$
B $- \sqrt { 3 }$
C $\frac { \sqrt { 3 } } { 2 }$
D $- \frac { \sqrt { 3 } } { 2 }$
E $\frac { \sqrt { 5 } } { 5 }$ F $- \frac { \sqrt { 5 } } { 5 }$ G $\frac { 2 \sqrt { 5 } } { 5 }$ H $- \frac { 2 \sqrt { 5 } } { 5 }$

tmua 2021 Q6 1 marks View

The function f is given by
$$\mathrm { f } ( x ) = \frac { \cos x + 3 } { 7 + 5 \cos x - \sin ^ { 2 } x }$$
Find the positive difference between the maximum and the minimum values of $\mathrm { f } ( x )$.
A 0 B $\frac { 1 } { 3 }$ C $\frac { 1 } { 2 }$ D $\frac { 2 } { 3 }$ E 1 F 2

turkey-yks 2011 Q27 View

Given that $0 < x < \frac { \pi } { 2 }$ and $\cot \mathrm { x } - 3 \tan \mathrm { x } = \frac { 1 } { \sin 2 \mathrm { x } }$, what is $\sin ^ { 2 } x$?
A) $\frac { 1 } { 9 }$
B) $\frac { 1 } { 8 }$
C) $\frac { 1 } { 7 }$

turkey-yks 2012 Q23 View

$$\frac { \cos 135 ^ { \circ } + \cos 330 ^ { \circ } } { \sin 150 ^ { \circ } }$$
What is the value of this expression?
A) $\sqrt { 3 } - \sqrt { 2 }$
B) $\sqrt { 3 } - 1$
C) $\sqrt { 2 } - 1$
D) $\sqrt { 2 } + 1$
E) $\sqrt { 2 } + \sqrt { 3 }$

turkey-yks 2013 Q24 View

$$\frac { \cot x } { \tan x + \cot x } = 4 \sin x - 3$$
Given this, what is the value of $\sin x$?
A) $3 - 2 \sqrt { 2 }$
B) $1 - \sqrt { 3 }$
C) $- 1 + \sqrt { 2 }$
D) $- 1 + \sqrt { 3 }$
E) $- 2 + 2 \sqrt { 2 }$

turkey-yks 2015 Q24 Find an angle using the cosine rule View

$| \mathrm { AB } | = 6$ units $| \mathrm { BH } | = 2$ units $[ \mathrm { BC } ] \cap [ \mathrm { GF } ] = \mathrm { H }$ $\mathrm { m } ( \widehat { \mathrm { GHC } } ) = \mathrm { x }$
In the figure, $ABCD$ and $AEFG$ are congruent squares. Accordingly, what is the value of $\tan ( x )$?
A) $\frac { 1 } { 3 }$
B) $\frac { 2 } { 3 }$
C) $\frac { 5 } { 3 }$
D) $\frac { 3 } { 4 }$
E) $\frac { 5 } { 4 }$

turkey-yks 2016 Q26 Find an angle using the cosine rule View

ABCD is a square, $\mathrm { AE } \cap \mathrm { BF } = \{ \mathrm { G } \}$, $| \mathrm { BC } | = 6$ units, $| \mathrm { DE } | = 4$ units, $| \mathrm { AF } | = 3$ units, $\mathrm { m } ( \widehat { \mathrm { FGE } } ) = \mathrm { x }$.
According to the given information above, what is the value of $\cot ( x )$?
A) $\frac { - 1 } { 4 }$
B) $\frac { - 5 } { 4 }$
C) $\frac { - 3 } { 8 }$
D) $\frac { - 1 } { 8 }$
E) $\frac { - 5 } { 8 }$

turkey-yks 2017 Q49 View

Given that $0 < x < \frac { \pi } { 2 }$, $$\frac { \sec ( x ) - 1 } { 2 } = \frac { 3 } { \sec ( x ) + 1 }$$ the equality holds.\ Accordingly, what is the value of $\tan ( x )$?\ A) $\sqrt { 2 }$\ B) $\sqrt { 3 }$\ C) $\sqrt { 5 }$\ D) $\sqrt { 6 }$\ E) $\sqrt { 7 }$

turkey-yks 2019 Q32 View

Let $0 < \mathrm { x } < \frac { \pi } { 2 }$. $\sec x \cdot \tan x \cdot ( 1 - \sin x ) = \frac { 1 } { 4 }$ Accordingly, what is the value of $\csc x$?
A) $\frac { 3 } { 2 }$
B) $\frac { 5 } { 2 }$
C) $\frac { 7 } { 2 }$
D) 2
E) 3

turkey-yks 2020 Q27 Trigonometric Identity Simplification View

$$\frac{2\tan x - \sin(2x)}{\sin^2 x}$$
What is the simplified form of this expression?
A) $2\tan x$
B) $\tan(2x)$
C) $2\cos x$
D) $\cos(2x)$
E) 1

turkey-yks 2023 Q31 Trigonometric Identity Simplification View

In the figure, a semicircle with center $O$ and diameter $AD$, a rectangle $ABCD$, and a triangle $OEF$ are given. Points $C$, $F$, $E$, $B$ are collinear; points $E$ and $F$ are on the circle.
Accordingly, what is the ratio of the area of rectangle $ABCD$ to the area of triangle $OEF$ in terms of $x$?
A) $\tan\frac{x}{2}$ B) $2 \cdot \sec x$ C) $2 \cdot \operatorname{cosec}\frac{x}{2}$ D) $2 \cdot \tan x$ E) $\cot x$

turkey-yks 2025 Q27 View

The simplified form of the expression
$$\frac{1 - \cos(4x)}{\sin(4x) + 2 \cdot \sin(2x)}$$
is which of the following?
A) $\sin x$ B) $\tan x$ C) $\cot x$ D) $\sec x$ E) $\operatorname{cosec} x$

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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

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