About Lesson
9.1 Basic Trigonometric Identities 基本三角关系式
1. Prove that 试证
(a)
(b)
*** QuickLaTeX cannot compile formula: \frac{1-tan^2x}{1+tan^2x} = 1-2 〖sin〗^2x *** Error message: Unicode character (U+2061) leading text: $\frac{1-tan^2x}{1+tan^2x}
(c) ![Rendered by QuickLaTeX.com \frac{sec x-cosecx }{tan x - cot x} = \frac{tan x + cot x}{sec x + cosec x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-97b4b59a01e0af42f97c09ed92441ee4_l3.png)
![Rendered by QuickLaTeX.com \frac{sec x-cosecx }{tan x - cot x} = \frac{tan x + cot x}{sec x + cosec x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-97b4b59a01e0af42f97c09ed92441ee4_l3.png)
(d) ![Rendered by QuickLaTeX.com \frac{{sin}^2{x}-{cos}^2{x}}{1+2sin{x}cos{x}}=\frac{tan{x}-1}{tan{x}+1}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-cb6f6843e40e56722b122d8389ba0bbb_l3.png)
![Rendered by QuickLaTeX.com \frac{{sin}^2{x}-{cos}^2{x}}{1+2sin{x}cos{x}}=\frac{tan{x}-1}{tan{x}+1}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-cb6f6843e40e56722b122d8389ba0bbb_l3.png)
(e) ![Rendered by QuickLaTeX.com \frac{cos{(}-x)-1}{sin{(}3\pi+x)}=\frac{1}{cos{e}c(\pi-x)-tan{(}x-\frac{3\pi}{2})}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-71f21ebfceabec03f81dbeffba3066be_l3.png)
![Rendered by QuickLaTeX.com \frac{cos{(}-x)-1}{sin{(}3\pi+x)}=\frac{1}{cos{e}c(\pi-x)-tan{(}x-\frac{3\pi}{2})}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-71f21ebfceabec03f81dbeffba3066be_l3.png)
(f) ![Rendered by QuickLaTeX.com \frac{2(cos{x}+1)}{sec{x}+tan{x}+1}=1-sin{x}+cos{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-7ee4c153543b325e5594477e6b74e10c_l3.png)
![Rendered by QuickLaTeX.com \frac{2(cos{x}+1)}{sec{x}+tan{x}+1}=1-sin{x}+cos{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-7ee4c153543b325e5594477e6b74e10c_l3.png)
(g) ![Rendered by QuickLaTeX.com \frac{tan{(}\pi+x)+cot{(}\frac{3\pi}{2}-x)}{1+tan{(}2\pi-x)cot{(}\frac{\pi}{2}+x)}=2sin{x}cos{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-dd8c0399ca4543e46345267ed9e5a623_l3.png)
![Rendered by QuickLaTeX.com \frac{tan{(}\pi+x)+cot{(}\frac{3\pi}{2}-x)}{1+tan{(}2\pi-x)cot{(}\frac{\pi}{2}+x)}=2sin{x}cos{x}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-dd8c0399ca4543e46345267ed9e5a623_l3.png)
(2)
If
, prove that
If
![Rendered by QuickLaTeX.com u=\frac{1+sin{x}}{cos{x}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-62025696a0ac815203ce22c9ffcaf3e9_l3.png)
![Rendered by QuickLaTeX.com \frac{1}{u}=\frac{1-sin{x}}{cos{x}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-d26a64607626d0ef334de62181a79fe2_l3.png)
若 , 证明
(3)
If
, show that
If
![Rendered by QuickLaTeX.com tan^2A - 2tan^2B = 1](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-f6a16ae6891337b8263bcba31fb7cff5_l3.png)
![Rendered by QuickLaTeX.com 2cos^2A - cos^2B = 0](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4a1718ea3633b7a23118786a1c8126f2_l3.png)
若 , 证明
(4)
If
, find the values of
若
, 求下列各式的值
(a) sin A cos A
(b) tan A + cot A
If
![Rendered by QuickLaTeX.com sin A + cos A = \frac{1}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-10b9535f9b8383444b9f73eb55867b7b_l3.png)
若
![Rendered by QuickLaTeX.com sin A + cos A = \frac{1}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-10b9535f9b8383444b9f73eb55867b7b_l3.png)
(a) sin A cos A
(b) tan A + cot A
Answer 答案:
(a)
![Rendered by QuickLaTeX.com -\frac{3}{8}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-9e00fcc868b3ee80ad676931bc5a820f_l3.png)
(b)
(5)
If
, then find the value of ![Rendered by QuickLaTeX.com tan2x + secx cscx + cot2x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c4dd3c974847ca66687d3ceee51dbc10_l3.png)
若
则求
之值
If
![Rendered by QuickLaTeX.com tan x + cot x = 4\frac{1}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4a935995e82272f2837b2ed5f627464f_l3.png)
![Rendered by QuickLaTeX.com tan2x + secx cscx + cot2x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c4dd3c974847ca66687d3ceee51dbc10_l3.png)
若
![Rendered by QuickLaTeX.com tan x + cot x = 4\frac{1}{2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4a935995e82272f2837b2ed5f627464f_l3.png)
![Rendered by QuickLaTeX.com tan2x + secx cscx + cot2x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c4dd3c974847ca66687d3ceee51dbc10_l3.png)
Answer 答案:
![Rendered by QuickLaTeX.com 22\frac{3}{4}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-e71a309b79075d3824b2bfcb7d2907dc_l3.png)
(6)
Given that
, find
, if ![Rendered by QuickLaTeX.com 0 < x < 45^{\circ}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-a163d410a477a79eab8dafd3555014fd_l3.png)
已知
, 求
, 当
Given that
![Rendered by QuickLaTeX.com sin x + cos x = 1.2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-7c583b53a3c3046d49b7cdc1ee588b52_l3.png)
![Rendered by QuickLaTeX.com sin x - cos x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4f0f71263ac474775b800329cb753213_l3.png)
![Rendered by QuickLaTeX.com 0 < x < 45^{\circ}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-a163d410a477a79eab8dafd3555014fd_l3.png)
已知
![Rendered by QuickLaTeX.com sin x + cos x = 1.2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-7c583b53a3c3046d49b7cdc1ee588b52_l3.png)
![Rendered by QuickLaTeX.com sin x - cos x](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-4f0f71263ac474775b800329cb753213_l3.png)
![Rendered by QuickLaTeX.com 0 < x < 45^{\circ}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-a163d410a477a79eab8dafd3555014fd_l3.png)
Answer 答案:
![Rendered by QuickLaTeX.com -\sqrt{0.56}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-9f55ddf3eec3f3e69541b8991ba6942e_l3.png)
(7)
Given that
,
, prove that
.
Given that
![Rendered by QuickLaTeX.com msec{\theta}=1+tan{\theta}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-bc4a3f1c7807b6cdfb49db2d32a58dec_l3.png)
![Rendered by QuickLaTeX.com nsec{\theta}=1-tan{\theta}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-c0dea1a65c6eb8648e96689770c254f2_l3.png)
![Rendered by QuickLaTeX.com m^2 + n^2 = 2](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-fc1b4a496c4888c853b21b371c6f7b68_l3.png)
已知 ,
, 证明
.
(8)
If
show that
If
![Rendered by QuickLaTeX.com \frac{a}{sin{A}}=\frac{b}{cos{A}}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-2d143139c23c068186f3e4b4302616b6_l3.png)
![Rendered by QuickLaTeX.com sin{A}cos{A}=\frac{ab}{a^2+b^2}](https://learn-ondemand.com/wp-content/ql-cache/quicklatex.com-1d0da66f065c6333e04106ad01dc9f2e_l3.png)
若 证明