第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

9.3 倍角及半角的三角函数 Trigonometric Functions of Multiple Angle and Half Angle

(1) Prove the following identities
证明下列恒等式

(a) \frac{cos{2}x}{sin{x}-cos{x}}=-cos{x}-sin{x}
(b) \frac{(sin{\frac{x}{2}}+cos{\frac{x}{2}})^2}{1+sin{x}}=1
(c) \frac{tan{2}x}{{tan}^2{2}x-1}=-\frac{1}{2}tan{4}x
(d) \frac{sin{3}x}{sin{x}}+\frac{cos{3}x}{cos{x}}=4cos{2}x
(e) \frac{sin{2}x}{1+cos{2}x}\cdot\frac{cos{x}}{1+cos{x}}=tan{\frac{x}{2}}
(f) \frac{sin{2}x+cos{2}x+1}{sin{2}x-cos{2}x+1}=cot{x}
(g) \frac{1+sin{x}}{1-sin{x}}={tan}^2{(}\frac{\pi}{4}+\frac{x}{2})
(h) {cos}^6{x}+{sin}^6{x}=1-\frac{3}{4}{sin}^2{2}x
(i) \frac{sin{2}A+cos{A}}{cos{2}A+sin{A}}=\frac{cos{A}}{1-sin{A}}
(j) \frac{sin{2}x}{(sin{x}+cos{x}-1)(sin{x}-cos{x}+1)}=cot{\frac{x}{2}}

(2)
If tan(A + B) = \frac{1}{7}, tan A = 3 and sin A < 0; find without using calculator or tables, the values of
tan(A + B) = \frac{1}{7}, tan A = 3sin A < 0; 不用计算机或查表,求
(a) tan B
(b) sin \frac{1}{2}A
(c) cos 2A

Answer 答案:
(a) -2
(b) \pm0.811
(c) -\frac{4}{5}
(3)
Given that 2sin^{2}x + 4sin x cos x + 5cos^{2}x \equiv a + b sin2x + c cos2x, find the values of a, b and c.
已知 2sin^{2}x + 4sin x cos x + 5cos^{2}x \equiv a + b sin2x + c cos2x, 求 a, b 和 c的值.
Answer 答案:

 3\frac{1}{2} ,  2 ,  1\frac{1}{2}