LFM Pure

csat-suneung 2008 Q26 3 marks Direct Double Angle Evaluation View

(Calculus) When $\sin \alpha = \frac { 3 } { 4 }$, what is the value of $\cos 2 \alpha$? [3 points]
(1) $- \frac { 1 } { 32 }$
(2) $- \frac { 1 } { 16 }$
(3) $- \frac { 1 } { 8 }$
(4) $- \frac { 1 } { 4 }$
(5) $- \frac { 1 } { 2 }$

csat-suneung 2010 Q26 3 marks Direct Double Angle Evaluation View

[Calculus] When $\tan \theta = - \sqrt { 2 }$, what is the value of $\sin \theta \tan 2 \theta$? (where $\frac { \pi } { 2 } < \theta < \pi$ ) [3 points]
(1) $\frac { 2 \sqrt { 3 } } { 3 }$
(2) $\sqrt { 3 }$
(3) $\frac { 4 \sqrt { 3 } } { 3 }$
(4) $\frac { 5 \sqrt { 3 } } { 3 }$
(5) $2 \sqrt { 3 }$

csat-suneung 2013 Q3 2 marks Direct Double Angle Evaluation View

When $\sin \theta = \frac { 1 } { 3 }$, what is the value of $\sin 2 \theta$? (Given that $0 < \theta < \frac { \pi } { 2 }$.) [2 points]
(1) $\frac { 7 \sqrt { 2 } } { 18 }$
(2) $\frac { 4 \sqrt { 2 } } { 9 }$
(3) $\frac { \sqrt { 2 } } { 2 }$
(4) $\frac { 5 \sqrt { 2 } } { 9 }$
(5) $\frac { 11 \sqrt { 2 } } { 18 }$

csat-suneung 2013 Q23 3 marks Function Analysis via Identity Transformation View

Find the maximum value $a$ of the function $f ( x ) = 2 \cos \left( x - \frac { \pi } { 3 } \right) + 2 \sqrt { 3 } \sin x$. Find the value of $a ^ { 2 }$. [3 points]

csat-suneung 2014 Q3 2 marks Direct Double Angle Evaluation View

When $\tan \theta = \frac { \sqrt { 5 } } { 5 }$, what is the value of $\cos 2 \theta$? [2 points]
(1) $\frac { \sqrt { 2 } } { 3 }$
(2) $\frac { \sqrt { 3 } } { 3 }$
(3) $\frac { 2 } { 3 }$
(4) $\frac { \sqrt { 5 } } { 3 }$
(5) $\frac { \sqrt { 6 } } { 3 }$

csat-suneung 2020 Q9 3 marks Addition/Subtraction Formula Evaluation View

In an isosceles triangle ABC with $\overline { \mathrm { AB } } = \overline { \mathrm { AC } }$, let $\angle \mathrm { A } = \alpha , \angle \mathrm { B } = \beta$. If $\tan ( \alpha + \beta ) = - \frac { 3 } { 2 }$, what is the value of $\tan \alpha$? [3 points]
(1) $\frac { 5 } { 2 }$
(2) $\frac { 12 } { 5 }$
(3) $\frac { 23 } { 10 }$
(4) $\frac { 11 } { 5 }$
(5) $\frac { 21 } { 10 }$

gaokao 2015 Q6 5 marks Addition/Subtraction Formula Evaluation View

If $\tan a = \frac { 1 } { 3 } , \tan ( a + b ) = \frac { 1 } { 2 }$, then $\tan b =$
(A) $\frac { 1 } { 7 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 5 } { 7 }$
(D) $\frac { 5 } { 6 }$

gaokao 2015 Q8 Addition/Subtraction Formula Evaluation View

8. Given $\tan \alpha = - 2 , \tan ( \alpha + \beta ) = \frac { 1 } { 7 }$, then the value of $\tan \beta$ is $\_\_\_\_$ .

gaokao 2015 Q9 Simplification of Trigonometric Expressions with Specific Angles View

9. If $\tan \alpha = 2 \tan \frac { \pi } { 5 }$, then $\frac { \cos \left( \alpha - \frac { 3 \pi } { 10 } \right) } { \sin \left( \alpha - \frac { \pi } { 5 } \right) } =$
A. $1$
B. $2$
C. $3$
D. $4$

gaokao 2015 Q12 Simplification of Trigonometric Expressions with Specific Angles View

12. $\sin 15 ^ { \circ } + \sin 75 ^ { \circ } = \_\_\_\_$.

gaokao 2015 Q14 Function Analysis via Identity Transformation View

14. Given the function $f ( x ) = \sin \omega x + \cos \omega x ( \omega > 0 ) , x \in \mathbb{R}$. If the function $f ( x )$ is monotonically increasing on the interval $( - \omega , \omega )$, and the graph of $f ( x )$ is symmetric about the line $x = \omega$, then the value of $\omega$ is $\_\_\_\_$.

III. Solution Questions: This section has 6 questions, for a total of 80 points.

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 Q16 14 marks Multi-Step Composite Problem Using Identities View

16. (14 points) In $\triangle A B C$ , the sides opposite to angles $\mathrm { A } , \mathrm { B }$ , C are $a , b , c$ respectively. Given that $\tan \left( \frac { \pi } { 4 } + \mathrm { A } \right) = 2$ .
(1) Find the value of $\frac { \sin 2 A } { \sin 2 A + \cos ^ { 2 } A }$ ;
(2) If $\mathrm { B } = \frac { \pi } { 4 } , a = 3$ , find the area of $\triangle A B C$ .

gaokao 2015 Q18 13 marks Function Analysis via Identity Transformation View

Given the function $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 2 } \sin 2 \mathrm { x } - \sqrt { 3 } \cos ^ { 2 } x$ .
(I) Find the minimum positive period and minimum value of $\mathrm { f } ( \mathrm { x } )$;
(II) The graph of function $\mathrm { f } ( \mathrm { x } )$ is transformed by stretching each point's horizontal coordinate to twice the original length while keeping the vertical coordinate unchanged, resulting in the graph of function $\mathrm { g } ( \mathrm { x } )$. When $\mathrm { x } \in \left[ \frac { \pi } { 2 } , \pi \right]$, find the range of $\mathrm { g } ( \mathrm { x } )$.

gaokao 2018 Q4 5 marks Direct Double Angle Evaluation View

If $\sin \alpha = \frac { 1 } { 3 }$, then $\cos 2 \alpha =$
A. $\frac { 8 } { 9 }$
B. $\frac { 7 } { 9 }$
C. $- \frac { 7 } { 9 }$
D. $- \frac { 8 } { 9 }$

gaokao 2018 Q11 5 marks Multi-Step Composite Problem Using Identities View

The vertex of angle $\alpha$ is at the origin, its initial side coincides with the positive $x$-axis, and two points on its terminal side are $A ( 1 , a )$ and $B ( 2 , b )$. If $\cos 2 \alpha = \frac { 2 } { 3 }$, then $| a - b | =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { 2 \sqrt { 5 } } { 5 }$
D. (incomplete)

gaokao 2018 Q15 5 marks Addition/Subtraction Formula Evaluation View

Given $\tan \left( \alpha - \frac { 5 \pi } { 4 } \right) = \frac { 1 } { 5 }$, then $\tan \alpha = $ \_\_\_\_ .

gaokao 2018 Q15 5 marks Trigonometric Equation Solving via Identities View

Given $\sin \alpha + \cos \beta = 1 , \cos \alpha + \sin \beta = 0$, then $\sin ( \alpha + \beta ) = \_\_\_\_$.

gaokao 2019 Q10 Trigonometric Equation Solving via Identities View

10. Given $\alpha \in \left( 0 , \frac { \pi } { 2 } \right) , 2 \sin 2 \alpha = \cos 2 \alpha + 1$, then $\sin \alpha =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { 2 \sqrt { 5 } } { 5 }$

gaokao 2019 Q11 Trigonometric Equation Solving via Identities View

11. Given $a \in \left( 0 , \frac { \pi } { 2 } \right) , 2 \sin 2 \alpha = \cos 2 \alpha + 1$, then $\sin \alpha =$
A. $\frac { 1 } { 5 }$
B. $\frac { \sqrt { 5 } } { 5 }$
C. $\frac { \sqrt { 3 } } { 3 }$
D. $\frac { 2 \sqrt { 5 } } { 5 }$

gaokao 2019 Q14 Addition/Subtraction Formula Evaluation View

14. Given $\tan \left( \alpha + \frac { \pi } { 4 } \right) = 6$, then $\tan \alpha = $ \_\_\_\_.

gaokao 2020 Q2 5 marks Qualitative Reasoning about Double Angle Signs or Inequalities View

If $\alpha$ is an angle in the fourth quadrant, then
A． $\cos 2 \alpha > 0$
B． $\cos 2 \alpha \leqslant 0$
C． $\sin 2 \alpha > 0$
D． $\sin 2 \alpha < 0$

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 Addition/Subtraction Formula Evaluation View

Given $2 \tan \theta - \tan \left( \theta + \frac { \pi } { 4 } \right) = 7$ , then $\tan \theta =$
A. $- 2$
B. $- 1$
C. $1$
D. $2$

gaokao 2021 Q9 Trigonometric Equation Solving via Identities View

9. If $a \in \left(0, \frac{\pi}{2}\right)$, $\tan 2a = \frac{\cos a}{2 - \sin a}$, then $\tan a =$
A. $\frac{\sqrt{15}}{15}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{\sqrt{5}}{3}$
D. $\frac{\sqrt{15}}{3}$

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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

III. Solution Questions: This section has 6 questions, for a total of 80 points.

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