Property 19
Conversion Formulas
\(\sin x+\sin y=2\sin\samefrac{x+y}{2}\cos\samefrac{x-y}{2}\)
\(\sin x-\sin y=2\sin\samefrac{x-y}{2}\cos\samefrac{x+y}{2}\)
\(\cos x+\cos y=2\cos\samefrac{x+y}{2}\cos\samefrac{x-y}{2}\)
\(\cos x-\cos y=-2\sin\samefrac{x+y}{2}\sin\samefrac{x-y}{2}\)
1.
\(\sin105^\circ+\sin15^\circ=?\)
2.
\(\cos15^\circ+\cos75^\circ=?\)
3.
\(\samefrac{\sin20^\circ+\sin70^\circ}{\cos85^\circ+\cos35^\circ}=?\)
4.
\(\samefrac{\cos4x+\cos2x}{\sin4x+\sin2x}=?\)
5.
\(\samefrac{\sin x-\sin4x+\sin7x}{\cos x-\cos4x+\cos7x}=?\)
Property 20
Inverse Conversion Formulas
\(\cos a\cos b=\samefrac12[\cos(a+b)+\cos(a-b)]\)
\(\sin a\sin b=\samefrac12[\cos(a-b)-\cos(a+b)]\)
\(\sin a\cos b=\samefrac12[\sin(a+b)+\sin(a-b)]\)
\(\cos a\sin b=\samefrac12[\sin(a+b)-\sin(a-b)]\)
1.
\(\cos\samefrac{3\pi}{8}\cos\samefrac\pi8=?\)
2.
\(4\sin70^\circ\cos50^\circ-2\cos70^\circ=?\)
3.
\(\cos10^\circ=a\)
\(\cos35^\circ\sin65^\circ-\samefrac14=?\)
4.
\(\samefrac{1+\cos40^\circ}{\cos55^\circ\cos35^\circ}=?\)
Property 21
Trigonometric Equation
\(\sin\alpha=\sin\beta\Rightarrow\alpha=\beta+2k\pi\) or \(\alpha=(\pi-\beta)+2k\pi\)
\(\cos\alpha=\cos\beta\Rightarrow\alpha=\pm\beta+2k\pi\)
\(\tan\alpha=\tan\beta\Rightarrow\alpha=\beta+k\pi\)
\(\cot\alpha=\cot\beta\Rightarrow\alpha=\beta+k\pi\)
For \(a\sin x+b\cos x=0\), divide by \(\cos x\): \(\tan x=-\samefrac ba\). For \(a\sin x+b\cos x=c\), compare \(a^2+b^2\) with \(c^2\) to determine the number of solutions.
1.
\(0\le x<360^\circ,\ \sin x=\samefrac12\)
\(\Rightarrow S.S.=?\)
2.
\(2\cos x=\sqrt3\)
\(\Rightarrow S.S.=?\)
3.
\(\tan x=-1\)
\(\Rightarrow S.S.=?\)
4.
\(0\le x<360^\circ,\ \tan x=\samefrac{\sqrt3}{3}\)
\(\Rightarrow S.S.=?\)
5.
\(0\le x<180^\circ,\ \cot(9x+30^\circ)=\cot(7x+70^\circ)\)
\(\Rightarrow S.S.=?\)
6.
\(0\le x<360^\circ,\ 2\cos^2x-3\cos x+1=0\)
\(\Rightarrow S.S.=?\)
7.
\(0\le x<180^\circ,\ \sin x=\sqrt3\cos x\)
\(\Rightarrow S.S.=?\)