Property 12
Sequence in Trigonometric Functions
Cosinus
\(\alpha<\beta\)
\(\cos\beta<\cos\alpha\)
In the I. section cosinus is the decreasing function.
\(\cos50^\circ<\cos20^\circ\)
Sinus
\(\alpha<\beta\)
\(\sin\alpha<\sin\beta\)
In the I. section sinus is the increasing function.
\(\sin40^\circ<\sin50^\circ\)
Tangent
\(\alpha<\beta\)
\(\tan\alpha<\tan\beta\)
In the I. section tangent is the increasing function
\(\tan50^\circ<\tan60^\circ\)
Cotangent
\(\alpha<\beta\)
\(\cot\beta<\cot\alpha\)
In the I. section cotangent is the decreasing function
\(\cot30^\circ<\cot20^\circ\)
Example
If angles are not in the I. section values are moved to I. section by applying the property 11. Functions are converted to either sinus/cosinus or tangent/cotangent.
\(\sin\longleftrightarrow\cos\quad\tan\longleftrightarrow\cot\)
\(\alpha+\beta=90^\circ\quad\sin\alpha=\cos\beta\quad\tan\alpha=\cot\beta\)
\(\begin{aligned}a&=\sin20^\circ\longrightarrow\sin20^\circ\\b&=\cos50^\circ\longrightarrow\sin40^\circ\\c&=\sin170^\circ=\sin(180^\circ-10^\circ)\longrightarrow\sin10^\circ\\d&=\cos320^\circ=\cos(270^\circ+50^\circ)\longrightarrow\sin50^\circ\end{aligned}\)
\(c<a<b<d\)
Comparison of Sinus and Tangent
a) (if the angles are same)
\(\sin\alpha<\tan\alpha\quad\left(0<\alpha\le\samefrac\pi2\right)\)
b) (if the angles are different)
\(-1\le\sin\alpha\le1\)
\(\tan45^\circ=1\)
\(a=\sin70^\circ\longrightarrow a<1\)
\(b=\tan50^\circ\longrightarrow b>1\)
\(a<b\)
1.
\(a=\sin20^\circ\)
\(b=\cos110^\circ\)
\(c=\tan150^\circ\)
\(d=\cot210^\circ\)
What are the signs of the expressions respectively?
2.
\(a=\cos60^\circ\)
\(b=\sin210^\circ\)
\(c=\tan150^\circ\)
\(d=\cot100^\circ\)
What are the signs of the expressions respectively?
3.
\(a=\cos(-120^\circ)\)
\(b=\sin(-120^\circ)\)
\(c=\tan(-120^\circ)\)
What are the signs of the expressions respectively?
4.
\(x=\sin20^\circ\)
\(y=\sin35^\circ\)
\(z=\cos40^\circ\)
\(\Rightarrow ?<?<?\)
5.
\(a=\cos30^\circ\)
\(b=\sin140^\circ\)
\(c=\tan140^\circ\)
\(\Rightarrow ?<?<?\)
6.
\(a=\cos124^\circ\)
\(b=\cos724^\circ\)
\(c=\cos744^\circ\)
\(\Rightarrow ?<?<?\)
7.
\(a=\cos140^\circ\)
\(b=\sin140^\circ\)
\(c=\tan130^\circ\)
\(\Rightarrow ?<?<?\)