Giải SBT Toán 11 Cánh Diều Các phép biến đổi lượng giác có đáp án

Đáp án đúng là: A

Ta có $\tan \left( {a + b} \right) = \frac{{\tan a + \tan b}}{{1 - \tan a.\tan b}} = \frac{{\frac{1}{7} + \frac{3}{4}}}{{1 - \frac{1}{7}.\frac{3}{4}}} = \frac{{\frac{{25}}{{28}}}}{{\frac{{25}}{{28}}}} = 1$.

Đáp án đúng là: A

Vì $0 < \alpha < \frac{\pi }{2}$ nên cos α > 0, do đó từ sin$^{2}$ α + cos$^{2}$ α = 1, suy ra

$\cos \alpha = \sqrt {1 - {{\sin }^2}\alpha } = \sqrt {1 - {{\left( {\frac{1}{{\sqrt 3 }}} \right)}^2}} = \frac{{\sqrt 6 }}{3}$.

Ta có $\cos \left( {\alpha + \frac{\pi }{3}} \right)$$= \cos \alpha \cos \frac{\pi }{3} - \sin \alpha \,\sin \frac{\pi }{3}$$= \frac{{\sqrt 6 }}{3}.\frac{1}{2} - \frac{1}{{\sqrt 3 }}.\frac{{\sqrt 3 }}{2} = \frac{{\sqrt 6 }}{6} - \frac{1}{2}$.

Đáp án đúng là: D

Ta có $P = \left( {1 - 3\cos 2\alpha } \right)\left( {2 + 3\cos 2\alpha } \right)$

$= \left[ {1 - 3\left( {1 - 2{{\sin }^2}\alpha } \right)} \right]\left[ {2 + 3\left( {1 - 2{{\sin }^2}\alpha } \right)} \right]$

$= \left( { - 2 + 6{{\sin }^2}\alpha } \right)\left( {5 - 6{{\sin }^2}\alpha } \right)$

$= \left[ { - 2 + 6.{{\left( {\frac{2}{3}} \right)}^2}} \right].\left[ {5 - 6.{{\left( {\frac{2}{3}} \right)}^2}} \right] = \frac{{14}}{9}$.

Đáp án đúng là: B

Ta có sin$^{4}$ x + cos$^{4}$ x = 1 – 2sin$^{2}$ x cos$^{2}$ x (theo Bài 9a)

= 1 – 2 (sin x cos x)$^{2}$ = $1 - 2{\left( {\frac{{\sin 2x}}{2}} \right)^2} = 1 - 2.\frac{{{{\sin }^2}2x}}{4} = 1 - \frac{{2\left( {1 - {{\cos }^2}2x} \right)}}{4}$

$= 1 - \frac{{2 - 2{{\cos }^2}2x}}{4} = \frac{{4 - 2 + 2{{\cos }^2}2x}}{4}$$= \frac{{3 + \left( {2{{\cos }^2}2x - 1} \right)}}{4} = \frac{{3 + \cos 4x}}{4}$.

Vậy ${\sin ^4}x + {\cos ^4}x = \frac{{3 + \cos 4x}}{4}$.

Đáp án đúng là: A

Ta có cos(120° – x) + cos(120° + x) – cos x

= cos 120° cos x + sin 120° sin x + cos 120° cos x – sin 120° sin x – cos x

= 2 cos 120° cos x – cos x

= 2 . $\left( { - \frac{1}{2}} \right)$ . cos x – cos x

= – cos x – cos x

= – 2 cos x.

Đáp án đúng là: B

Ta có $\cos \frac{a}{2}\cos \frac{a}{2} = {\cos ^2}\frac{a}{2} = \frac{{1 + \cos 2.\frac{a}{2}}}{2} = \frac{{1 + \cos a}}{2} = \frac{{1 + \frac{3}{4}}}{2} = \frac{7}{8}$.

Đáp án đúng là: A

Ta có $A = 4\sin \left( {a + \frac{\pi }{3}} \right)\sin \left( {a - \frac{\pi }{3}} \right)$

$= 4.\left( { - \frac{1}{2}} \right).\left[ {\cos \left( {a + \frac{\pi }{3} + a - \frac{\pi }{3}} \right) - \cos \left( {a + \frac{\pi }{3} - a + \frac{\pi }{3}} \right)} \right]$

$= - 2\left( {\cos 2a - \cos \frac{{2\pi }}{3}} \right)$

$= - 2\left[ {\left( {2{{\cos }^2}a - 1} \right) - \left( { - \frac{1}{2}} \right)} \right]$

$= - 2\left[ {2.{{\left( {\frac{{\sqrt 5 }}{3}} \right)}^2} - 1 + \frac{1}{2}} \right] = - \frac{{11}}{9}$.

Đáp án đúng là: D

Ta có cos(a + b) cos(a − b) $= \frac{1}{2}\left[ {\cos \left( {a + b + a - b} \right) + \cos \left( {a + b - a + b} \right)} \right]$

$= \frac{1}{2}\left( {\cos 2a + \cos 2b} \right)$

$= \frac{1}{2}\left[ {\left( {2{{\cos }^2}a - 1} \right) + \left( {1 - 2{{\sin }^2}b} \right)} \right]$

$= \frac{1}{2}.\left[ {2.{{\left( {\frac{1}{3}} \right)}^2} - 2.{{\left( { - \frac{2}{3}} \right)}^2}} \right] = - \frac{1}{3}$.

Đáp án đúng là: C

Ta có $P = \frac{{\sin \frac{\pi }{9} + \sin \frac{{5\pi }}{9}}}{{\cos \frac{\pi }{9} + \cos \frac{{5\pi }}{9}}}$$= \frac{{2\sin \frac{{\frac{\pi }{9} + \frac{{5\pi }}{9}}}{2}\cos \frac{{\frac{\pi }{9} - \frac{{5\pi }}{9}}}{2}}}{{2\cos \frac{{\frac{\pi }{9} + \frac{{5\pi }}{9}}}{2}\cos \frac{{\frac{\pi }{9} - \frac{{5\pi }}{9}}}{2}}}$

$= \frac{{\sin \frac{\pi }{3}\cos \left( { - \frac{{2\pi }}{9}} \right)}}{{\cos \frac{\pi }{3}\cos \left( { - \frac{{2\pi }}{9}} \right)}} = \frac{{\sin \frac{\pi }{3}}}{{\cos \frac{\pi }{3}}} = \frac{{\frac{{\sqrt 3 }}{2}}}{{\frac{1}{2}}} = \sqrt 3$.

Đáp án đúng là: C

Ta có $A = \frac{{\sin x + \sin 2x + \sin 3x}}{{\cos x + \cos 2x + \cos 3x}}$$= \frac{{\left( {\sin x + \sin 3x} \right) + \sin 2x}}{{\left( {\cos x + \cos 3x} \right) + \cos 2x}}$

$= \frac{{2\sin \frac{{x + 3x}}{2}\cos \frac{{x - 3x}}{2} + \sin 2x}}{{2\cos \frac{{x + 3x}}{2}\cos \frac{{x - 3x}}{2} + \cos 2x}}$$= \frac{{2\sin 2x\cos \left( { - x} \right) + \sin 2x}}{{2\cos 2x\cos \left( { - x} \right) + \cos 2x}}$

$= \frac{{\sin 2x\left( {2\cos x + 1} \right)}}{{\cos 2x\left( {2\cos x + 1} \right)}} = \frac{{\sin 2x}}{{\cos 2x}} = \tan 2x$.

Vì $\frac{\pi }{2} < a < \pi$ nên cos a < 0, do đó từ sin$^{2}$ a + cos$^{2}$ a = 1, suy ra

$\cos a = - \sqrt {1 - {{\sin }^2}a} = - \sqrt {1 - {{\left( {\frac{2}{3}} \right)}^2}} = - \frac{{\sqrt 5 }}{3}$.

Ta có $\tan a = \frac{{\sin a}}{{\cos a}} = \frac{{\frac{2}{3}}}{{ - \frac{{\sqrt 5 }}{3}}} = - \frac{{2\sqrt 5 }}{5}$.

$\sin \left( {a + \frac{\pi }{4}} \right)$$= \sin a\cos \frac{\pi }{4} + \cos a\sin \frac{\pi }{4}$$= \frac{2}{3}.\frac{{\sqrt 2 }}{2} + \left( { - \frac{{\sqrt 5 }}{3}} \right).\frac{{\sqrt 2 }}{2} = \frac{{2\sqrt 2 - \sqrt {10} }}{6}$.

$\cos \left( {a - \frac{{5\pi }}{6}} \right)$$= \cos a\cos \frac{{5\pi }}{6} + \sin a\sin \frac{{5\pi }}{6}$$= \left( { - \frac{{\sqrt 5 }}{3}} \right).\left( { - \frac{{\sqrt 3 }}{2}} \right) + \frac{2}{3}.\frac{1}{2} = \frac{{\sqrt {15} + 2}}{6}$.

$\tan \left( {a + \frac{{2\pi }}{3}} \right)$$= \frac{{\tan a + \tan \frac{{2\pi }}{3}}}{{1 - \tan a\tan \frac{{2\pi }}{3}}} = \frac{{ - \frac{{2\sqrt 5 }}{5} + \left( { - \sqrt 3 } \right)}}{{1 - \left( { - \frac{{2\sqrt 5 }}{5}} \right).\left( { - \sqrt 3 } \right)}} = \frac{{8\sqrt 5 + 9\sqrt 3 }}{7}$.

$\sin 2a = 2\sin a\cos a = 2.\frac{2}{3}.\left( { - \frac{{\sqrt 5 }}{3}} \right) = - \frac{{4\sqrt 5 }}{9}$.

$\cos 2a = 2{\cos ^2}a - 1 = 2.{\left( { - \frac{{\sqrt 5 }}{3}} \right)^2} - 1 = \frac{1}{9}$.

Do π < a < 2π nên $\frac{\pi }{2} < \frac{a}{2} < \pi$. Suy ra $\sin \frac{a}{2} > 0,\,\,\cos \frac{a}{2} < 0$.

Ta có: ${\sin ^2}\frac{a}{2} = \frac{{1 - \cos a}}{2} = \frac{{1 - 0{,}2}}{2} = 0{,}4$, suy ra $\sin \frac{a}{2} = \frac{{\sqrt {10} }}{5}$.

Do đó, $\cos \frac{a}{2} = - \sqrt {1 - {{\sin }^2}\frac{a}{2}} = - \sqrt {1 - {{\left( {\frac{{\sqrt {10} }}{5}} \right)}^2}} = - \frac{{\sqrt {15} }}{5}$.

$\tan \frac{a}{2} = \frac{{\sin \frac{a}{2}}}{{\cos \frac{a}{2}}} = \frac{{\frac{{\sqrt {10} }}{5}}}{{ - \frac{{\sqrt {15} }}{5}}} = - \frac{{\sqrt 6 }}{3}$.

Ta có $\sin a = 2\sin \frac{a}{2}\cos \frac{a}{2} = \frac{{2\sin \frac{a}{2}\cos \frac{a}{2}}}{{{{\sin }^2}\frac{a}{2} + {{\cos }^2}\frac{a}{2}}}$ (do ${\sin ^2}\frac{a}{2} + {\cos ^2}\frac{a}{2} = 1$)

$= \frac{{2\tan \frac{a}{2}}}{{{{\tan }^2}\frac{a}{2} + 1}} = \frac{{2.\frac{1}{{\sqrt 2 }}}}{{{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2} + 1}} = \frac{{2\sqrt 2 }}{3}$.

$\cos a = {\cos ^2}\frac{a}{2} - {\sin ^2}\frac{a}{2} = \frac{{{{\cos }^2}\frac{a}{2} - {{\sin }^2}\frac{a}{2}}}{{{{\sin }^2}\frac{a}{2} + {{\cos }^2}\frac{a}{2}}}$$= \frac{{1 - {{\tan }^2}\frac{a}{2}}}{{{{\tan }^2}\frac{a}{2} + 1}} = \frac{{1 - {{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2}}}{{{{\left( {\frac{1}{{\sqrt 2 }}} \right)}^2} + 1}} = \frac{1}{3}$.

$\tan a = \frac{{\sin a}}{{\cos a}} = \frac{{\frac{{2\sqrt 2 }}{3}}}{{\frac{1}{3}}} = 2\sqrt 2$.

Ta có cos(a + 2b) = 2cos a

⇔ cos[(a + b) + b] = 2cos[(a + b) – b]

⇔ cos(a + b) . cos b – sin(a + b) . sin b = 2[cos(a + b) . cos b + sin(a + b) . sin b]

⇔ cos(a + b) . cos b – 2 cos(a + b) . cos b = 2 sin(a + b) . sin b + sin(a + b) . sin b

⇔ – cos(a + b) . cos b = 3 sin(a + b) . sin b

⇔ sin(a + b) . sin b = $- \frac{1}{3}$ cos(a + b) . cos b

$\Leftrightarrow \frac{{\sin \left( {a + b} \right)\sin b}}{{\cos \left( {a + b} \right)\cos b}} = - \frac{1}{3}$

⇔ tan(a + b) tan b = $\frac{{ - 1}}{3}$.

Vì tam giác ABC không vuông nên A, B, C khác $\frac{\pi }{2}$, do đó tan A, tan B, tan C xác định.

Do A + B + C = π nên A + B = π – C, do đó tan(A + B) = tan(π – C) = tan(– C) = – tanC.

Mà $\tan \left( {A + B} \right) = \frac{{\tan A + \tan B}}{{1 - \tan A\tan B}}$.

Khi đó $\frac{{\tan A + \tan B}}{{1 - \tan A\,\tan B}} = - \tan C$

⇔ tan A + tan B = – tan C . (1 – tan A . tan B)

⇔ tan A + tan B = – tan C + tan A . tan B . tan C

⇔ tan A + tan B + tan C = tan A . tan B . tan C.

Ta có $\frac{{A + B + C}}{2} = \frac{\pi }{2}$, suy ra $\frac{A}{2} + \frac{B}{2} = \frac{\pi }{2} - \frac{C}{2}$ nên $\tan \left( {\frac{A}{2} + \frac{B}{2}} \right) = \cot \frac{C}{2}$

$\Leftrightarrow \frac{{\tan \frac{A}{2} + \tan \frac{B}{2}}}{{1 - \tan \frac{A}{2}.\tan \frac{B}{2}}} = \frac{1}{{\tan \frac{C}{2}}}$

$\Leftrightarrow \left( {\tan \frac{A}{2} + \tan \frac{B}{2}} \right)\tan \frac{C}{2} = 1 - \tan \frac{A}{2}.\tan \frac{B}{2}$

$\Leftrightarrow \tan \frac{A}{2}.\tan \frac{C}{2} + \tan \frac{B}{2}.\tan \frac{C}{2} + \tan \frac{A}{2}.\tan \frac{B}{2} = 1$

$\Leftrightarrow \tan \frac{A}{2}.\tan \frac{B}{2} + \tan \frac{B}{2}.\tan \frac{C}{2} + \tan \frac{C}{2}.\tan \frac{A}{2} = 1$.

Trong tam giác vuông ABM, có $\tan \widehat {BAM} = \frac{{BM}}{{BA}} = \frac{1}{2}$.

Trong tam giác vuông ADN, có $\tan \widehat {DAN} = \frac{{DN}}{{AD}} = \frac{{DN}}{{DC}} = \frac{1}{3}$.

Do đó, $\tan \left( {\widehat {BAM} + \widehat {DAN}} \right)$$= \frac{{\tan \widehat {BAM} + \tan \widehat {DAN}}}{{1 - \tan \widehat {BAM}.\tan \widehat {DAN}}}$$= \frac{{\frac{1}{2} + \frac{1}{3}}}{{1 - \frac{1}{2}.\frac{1}{3}}} = 1$.

Từ câu a) ta có $\tan \left( {\widehat {BAM} + \widehat {DAN}} \right)$ = 1 nên $\widehat {BAM} + \widehat {DAN} = 45^\circ$.

Suy ra $\widehat {MAN} = \widehat {BAD} - \left( {\widehat {BAM} + \widehat {DAN}} \right) = 90^\circ - 45^\circ = 45^\circ$.

Vậy góc chiếu sáng của đèn pin bằng 45°.