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第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

第十三章: 方程组 Simultaneous Equations

第十五章:二元一次不等式及线性规划 linear inequality in two variables and linear programming

9.5 三角函数的和差化积 Product Formulae II
(1) Prove that 证明

(a) cosx + 2cos3x + cos5x$ = $4cos^2x\cos3x

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

(c) sin^2x + sin^2(120° + x)+sin^2(120°- x) = \frac{1}{3}

(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}}

(c) \frac{sin{B}+sin{C}-sin{A}}{sin{A}+sin{B}+sin{C}} = tan{\frac{B}{2}}tan{\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}

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}tan (A - B) = \frac{3}{4}, 且 0^{\circ} < A + B < 180^{\circ}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}}

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)

sinA - sinB = \frac{1}{2}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)

(c) \cos(x-y)

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

(b) \frac{5}{13}

(c) -\frac{59}{72}

(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}}

{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}