9.5 三角函数的和差化积 Product Formulae II - 独中课程 | 线上补习

高中一 | 高级数学

About Lesson

(1) Prove that 证明

(a)

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(b) $\frac{sin{x}+sin{3}x+sin{5}x}{cos{x}+cos{3}x+cos{5}x}=tan{3}x$

(d) $tan(x - y) + tan(y - z) + tan( z - x )$ = $tan( x - y ) tan( y - z ) tan( z - x)$

(e) $cos^2x + cos^2y + cos^2z + cos^2( x + y + z)$ = $2 + 2cos( x + y) cos( y + z ) cos( z + x )$

(2) In triangle ABC, prove that 在三角形ABC中，证明

(a) $sin{A}+sin{B}+sin{C}$ = $4cos{\frac{A}{2}}cos{\frac{B}{2}}cos{\frac{C}{2}}$

(b) ${sin}^2{\frac{A}{2}}+{sin}^2{\frac{B}{2}}+{sin}^2{\frac{C}{2}}$ = $1-2sin{\frac{A}{2}}sin{\frac{B}{2}}sin{\frac{C}{2}}$

(3) Without using any calculator or tables, find the value of
不用计算机或查表，求下列各式的值

(a) $cos{\frac{\pi}{9}}-cos{\frac{2\pi}{9}}-cos{\frac{3\pi}{9}}-cos{\frac{4\pi}{9}}$

(b) $cos^2(30^{\circ} - A) + cos^2(30^{\circ} + A) - cos^2A$

Answer

(a) $-\frac{1}{2}$

(b) $\frac{1}{2}$

(4) If $sin (A + B) = k sin (A - B)$ , show that $tan{A}=\frac{k+1}{k-1}tan{B}$

若 $sin (A + B) = k sin (A - B)$ , 试证 $tan{A}=\frac{k+1}{k-1}tan{B}$

(5)
Prove that $\frac{sin{(}A+B)-sin{(}A-B)}{cos{(}A+B)+cos{(}A-B)}=tan{B}$ . If $tan (A + B) = -\frac{5}{12}$ and $tan (A - B) = \frac{3}{4}$ , where $0^{\circ} < A + B < 180^{\circ}$ and $0^{\circ} < A - B < 90^{\circ}$ . Find $tanB$

证明 $\frac{sin{(}A+B)-sin{(}A-B)}{cos{(}A+B)+cos{(}A-B)}=tan{B}$ . 若 $tan (A + B) = -\frac{5}{12}$ 和 $tan (A - B) = \frac{3}{4}$ , 且 $0^{\circ} < A + B < 180^{\circ}$ 及 $0^{\circ} < A - B < 90^{\circ}$ . 求 $tanB$

Answer

$1\frac{3}{4}$

(6) If $A + B + C = \frac{\pi}{2}$ , prove that $cos{A}+cos{B}+cos{C}$ = $4cos{\frac{B+C}{2}}cos{\frac{A+C}{2}}cos{\frac{A+B}{2}}$

若 $A + B + C = \frac{\pi}{2}$ , 试证 $cos{A}+cos{B}+cos{C}$ = $4cos{\frac{B+C}{2}}cos{\frac{A+C}{2}}cos{\frac{A+B}{2}}$

(7) If $sinA - sinB = \frac{1}{2}$ and $cosA - cosB = -\frac{1}{3}$ , find the value of $sin(A + B)$

若 $sinA - sinB = \frac{1}{2}$ 及 $cosA - cosB = -\frac{1}{3}$ , 求 $sin(A + B)$ 的值。

Answer

$\frac{12}{13}$

(8) Given that $sin x + sin y = -\frac{1}{3}$ , $cosx + cosy = \frac{1}{2}$ , find the value of

已知 $sin x + sin y = -\frac{1}{3}$ , $cosx + cosy = \frac{1}{2}$ , 求下列各式的值

(a) $\sin(x+y)$

(b) $\cos(x+y)$

Answer

(a) $\frac{12}{13}$

(b) $\frac{5}{13}$

(9) Find the value of ${cos}^4{\frac{\pi}{8}}$ + ${cos}^4{\frac{3\pi}{8}}$ + ${cos}^4{\frac{5\pi}{8}}$ + ${cos}^4{\frac{7\pi}{8}}$

求 ${cos}^4{\frac{\pi}{8}}$ + ${cos}^4{\frac{3\pi}{8}}$ + ${cos}^4{\frac{5\pi}{8}}$ + ${cos}^4{\frac{7\pi}{8}}$ 的值。

Answer

1 $\frac{1}{2}$