Property 9
Definition of Trigonometric Functions (Unit Circle)
Equation of a Unit Circle
\(x^2+y^2=1\)
\(\cos\alpha=a\)
\(\sin\alpha=b\)
\(\tan\alpha=c\)
\(\cot\alpha=d\)
Trigonometric Values of 0°, 90°, 180° and 270°
\(\begin{array}{cc}(x,&y)\\\downarrow&\downarrow\\\cos\alpha&\sin\alpha\end{array}\)
\(-1\le\cos\alpha\le1\)
\(-1\le\sin\alpha\le1\)
\(\begin{array}{ll}\cos0^\circ=1&\sin0^\circ=0\\\cos90^\circ=0&\sin90^\circ=1\\\cos180^\circ=-1&\sin180^\circ=0\\\cos270^\circ=0&\sin270^\circ=-1\end{array}\)
\(\tan0^\circ=0\)
\(\tan90^\circ=\text{undefined}\)
\(\tan180^\circ=0\)
\(\tan270^\circ=\text{undefined}\)
\(\cot0^\circ=\text{undefined}\)
\(\cot90^\circ=0\)
\(\cot180^\circ=\text{undefined}\)
\(\cot270^\circ=0\)
1.
\(\samefrac{\sin45^\circ+\tan45^\circ}{\cos180^\circ}=?\)
2.
\(\samefrac{\sin90^\circ-\cos90^\circ}{\cos0^\circ}=?\)
3.
\(\samefrac{\tan45^\circ-\sin180^\circ}{\cos60^\circ}=?\)
4.
\(\samefrac{\sin270^\circ+\sin60^\circ}{\cos45^\circ}=?\)
Property 10
Trigonometric Sections
\(\begin{array}{cc}0<\alpha<\samefrac\pi2&\samefrac\pi2<\alpha<\pi\\\pi<\alpha<\samefrac{3\pi}2&\samefrac{3\pi}2<\alpha<2\pi\end{array}\)
When a trigonometric value of an angle is given the numerical value of the other trigonometric values can be found in the right angle. The sign of the trigonometric function is given according to the section of an angle.
Example
\(\pi<\alpha<\samefrac{3\pi}2\)
If tan α = 2 what are the values of cos α, sin α and cot α?
Answer
\(\cot\alpha=\samefrac12\)
\(\cos\alpha=-\samefrac1{\sqrt5}\)
\(\sin\alpha=-\samefrac2{\sqrt5}\)
1.
\(90^\circ<x<180^\circ\)
\(\cos x=-\samefrac35\)
\(\tan x=?\)
2.
\(180^\circ<x<270^\circ\)
\(\tan x=\samefrac{15}{8}\)
\(\cos x=?\)
3.
\(270^\circ<x<360^\circ\)
\(\cos x=\samefrac{12}{13}\)
\(\tan x-\cot x=?\)
4.
\(\samefrac\pi2<x<\pi\)
\(\sin x=\samefrac13\)
\(\tan x-\cot x=?\)
5.
\(\samefrac\pi2<x<\pi\)
\(\cos x=-\samefrac45\)
\(\tan x+\cot x=?\)
6.
\(\samefrac{3\pi}{2}<x<2\pi\)
\(\tan x=-\samefrac23\)
\(\cos x-\sin x=?\)
7.
\(\pi<x<\samefrac{3\pi}{2}\)
\(\samefrac{\sin x-\cos x}{\sin x+\cos x}=\samefrac13\)
\(\cos x=?\)
8.
\(\pi<x<\samefrac{3\pi}{2}\)
\(2\sin x=5\cos x\)
\(\cot x=?\)