Property 11
While finding the trigonometric values of an obtuse angles 3 points below should be taken into consideration.
Obtuse angles are written by the help of 90°, 180°, 270° and 360°.
The sign is found according to the section.
If it is written by the help of 90° and 270° the trigonometric function changes.
\(\sin\alpha\longleftrightarrow\cos\alpha\)
\(\tan\alpha\longleftrightarrow\cot\alpha\)
If it is written by the help of 180° and 360° the trigonometric function does not change.
Example
\(\begin{aligned}\sin(210^\circ)&=\sin(180^\circ+30^\circ)=-\sin30^\circ\\&=\sin(270^\circ-60^\circ)=-\cos60^\circ\end{aligned}\)
\(\begin{aligned}\tan(120^\circ)&=\tan(180^\circ-60^\circ)=-\tan60^\circ\\&=\tan(90^\circ+30^\circ)=-\cot30^\circ\end{aligned}\)
Note
\(\blacksquare\quad \alpha+\beta=90^\circ\)
\(\sin\alpha=\cos\beta\)
\(\tan\alpha=\cot\beta\)
\(\blacksquare\quad \alpha+\beta=180^\circ\)
\(\cos\alpha=-\cos\beta\)
\(\sin\alpha=\sin\beta\)
\(\tan\alpha=-\tan\beta\)
\(\cot\alpha=-\cot\beta\)
\(\blacksquare\quad \cos(-\alpha)=\cos\alpha\)
\(\sin(-\alpha)=-\sin\alpha\)
\(\tan(-\alpha)=-\tan\alpha\)
\(\cot(-\alpha)=-\cot\alpha\)
1.
\(2\cdot\sin150^\circ+\cos120^\circ=?\)
2.
\(\tan225^\circ-\tan45^\circ=?\)
3.
\(\cos210^\circ+\tan300^\circ=?\)
4.
\(\samefrac{\tan225^\circ-\cot315^\circ}{\sin150^\circ}=?\)
5.
\(x=\sin25^\circ\)
\(\samefrac{\sin205^\circ+\cos115^\circ}{\tan(-25^\circ)}=?\)
6.
\(\samefrac{\cos170^\circ-\sin100^\circ}{\cos350^\circ}=?\)
7.
\(x=\cos20^\circ\)
\(\Rightarrow\samefrac{\cos(-20^\circ)}{\sin(-20^\circ)\cdot\cos160^\circ}=?\)
8.
\(x=\sin15^\circ\)
\(\cos105^\circ-\sin165^\circ=?\)
9.
\(0<x<90^\circ\)
\(\sin x=\samefrac12\)
\(\sin(90^\circ+x)-\tan(180^\circ-4x)=?\)
10.
\(0<x<\samefrac\pi2\)
\(\sin x=\samefrac23\)
\(\sin\left(\samefrac{3\pi}{2}-x\right)+\cos\left(\samefrac\pi2+x\right)=?\)
11.
\(0<\alpha<\samefrac\pi2\)
\(\cos(\pi-\alpha)+\sin\left(\samefrac{3\pi}{2}-\alpha\right)=?\)
12.
\(0<x<\samefrac\pi2\)
\(\tan\left(\samefrac\pi2+x\right)-\cot\left(\samefrac{3\pi}{2}+x\right)=?\)
13.
\(0<\alpha<\samefrac\pi2\)
\(\samefrac{\sin(9\pi-\alpha)-\cos\left(\samefrac{11\pi}{2}-\alpha\right)}{\tan(9\pi+\alpha)+\cot\left(\samefrac{3\pi}{2}-\alpha\right)}=?\)
14.
\(\pi<\alpha<\samefrac{3\pi}{2}\)
\(\cos(13\pi-\alpha)-\sin\left(\samefrac\pi2-\alpha\right)=1\)
\(\tan\alpha=?\)
15.
\(\sin(\alpha-7\pi)-\cos\left(\alpha-\samefrac{3\pi}{2}\right)=?\)
16.
\(\alpha+\beta=\samefrac{3\pi}{2}\)
\(\tan\alpha=\samefrac12\)
\(\sin(2\alpha+\beta)=?\)
17.
\(\alpha+\beta=90^\circ\)
\(\sin\alpha=\samefrac35\)
\(\tan(3\alpha+2\beta)-\cos(\beta-\pi)=?\)
18.
\(\alpha+\beta=\samefrac\pi2\)
\(\sin^2(2\beta+\alpha)-\cos^2(2\alpha+\beta)=?\)