第四章:部分分式 Partial Fraction

第六章:角的形成及单位 Angles and Measurements

9.2 两角和与差的三角函数 Compound Angles
(1)
Given that sin{A}=\frac{2}{3}, where \frac{\pi}{2}<A<\pi; cos{B}=-\frac{3}{4}, where \pi<B<\frac{3\pi}{2}, find the value of cos({A} - {B}).
已知 sin{A}=\frac{2}{3} , 且\frac{\pi}{2}<A<\pi ; , 且cos{B}=-\frac{3}{4} , 求 cos({A} - {B}) 的值.
Answer:
\frac{3\sqrt5-2\sqrt7}{12}
(2)
Prove that tan{7}5^{\circ}=2+3
证明tan{7}5^{\circ}=2+3
(3)
Show that, if A + B = \frac{\pi}{4}, cot A - cot A tan B - tan B = 1
试证,若 A + B = \frac{\pi}{4}, cot A - cot A tan B - tan B = 1
(4)
If tan x and tan y are the roots of equation x^2 - 7x  - 5 = 0, find the value of tan(x + y).
tan xtan y 为方程式x^2 - 7x  - 5 = 0的根, 求tan(x + y)的值.
Answer:
\frac{7}{6}
(5)
If A + B = 90^{\cirl} and A > 0^{\cirl}, B > 0^{\cirl}, prove that tan A tan B = 1.
Hence without using calculator or mathematical tables, find the value of tan 75^{\cirl} – tan15^{\cirl}.
若A + B = 90^{\cirl} 和 A > 0^{\cirl}, B > 0^{\cirl}, 证明 tan A tan B = 1.
据此不用计算机或查表,求 tan 75^{\cirl} – tan15^{\cirl} 的值.
Answer:
2\sqrt{3}
Prove the following identities.
证明下列恒等式
(6)
\frac{sin(A+B)}{cos{A}cos{B}}=tan{A}+tan{B}
(7)
\frac{sin(A-B)}{sin{A}sin{B}}=cot{B}-cot{A}
(8)
cos (A + B) cos (A - B) = cos^2A - sin^2B
(9)
sin (A + B) sin (A - B) = cos^2B - cos^2A
(10)
cos (45^{\circ} - A) - sin (45^{\circ} + A) = 0
(11)
cos (A - B) - sin (A + B) = (cosA -  sinA)(cosB - sinB)
(12)
cos (A + B) cosB + sin (A + B) sinB = cosA
(13)
sin3A cosA - cos3A sinA = sin2A
(14)
cos (30^{\circ} + A) cos(30^{\circ} - A) - sin(30^{\circ} + A) sin(30^{\circ} - A) = \frac{1}{2}
(15)
tan{(}A+B)-tan{A}=\frac{sin{B}}{cos{A}cos{(}A+B)}
(16)
Prove that in triangle ABC, tan A + tan B + tan C = tan A tan B tan C
试证在三角形ABC中, tan A + tan B + tan C = tan A tan B tan C