Evaluate the $\sin$, $\cos$ and $\tan$ without using calculator?
$150$ degree
the right answer are $\frac{1}{2}$, $-\frac{\sqrt{3}}{2}$and $-\frac{1}{\sqrt{3}} $
$-315$ degree
the right answer are $\frac{1}{\sqrt{2}}$, $\frac{1}{\sqrt{2}}$ and $1$.
$\endgroup$ 15 Answers
$\begingroup$Hints:
Write $150$ as $90 + 60$ and use $\sin(A+B),\cos(A+B),\tan(A+B)$ formulas.
Write $315$ as $270 + 45$
You can look up cos and sin on the unit circle.
The angles labelled above are those of the special right triangles 30-60-90 and 45-45-90. Note that -315 ≡ 45 (mod 360).
For tan, use the identity $\tan{\theta} = \frac{\sin{\theta}}{\cos \theta}$.
$\endgroup$ $\begingroup$It is very simple.
$\sin(150) = \sin(90+60) = \cos(60) = \frac{1}{2}$
$\cos(150) = \cos(90+60) = - \sin(60) = -\frac{\sqrt{3}}{2}$
$\tan(150) = \tan(90+60) = - \cot(60) = -\frac{1}{\sqrt{3}} $
similarly
$\sin(-315)=-\sin(270+45)=\cos(45) = \frac{1}{\sqrt{2}}$
$\cos(-315)=\cos(315)=\cos(270+45)=\sin(45) = \frac{1}{\sqrt{2}}$
$\tan(-315)=-\tan(270+45)=\cot(45) = 1$
Check this link for more info on converting trigo identities
$\endgroup$ $\begingroup$I would write $150=180-30$ and use $\cos (180-\theta)=-\cos \theta$ and so on. Then $-315=45-360$ so it is just the functions at $45$
$\endgroup$ $\begingroup$Two more possibilities:
- Use a table of trigonometric values.
- Use a ruler and a protractor in drawing the line values of the trigonometric functions. (See, for example, this.)
