Find the exact value of the six trigonometric functions of 285° using a trig identity

Question

I need help using the sum identity I have tried to use reference angles but I don't know how to start because the question just says 285° and not, for example sin285°

Answer 1

Hint: The "six trigonometric functions" means: $$\sin\quad \cos\quad\tan\\ \csc\quad\sec\quad\cot$$

Answer 2

$\text{Hint: }285^{\circ}=4(60^{\circ})+45^{\circ}$

$\text{Or you could use: }285^{\circ}=180^{\circ}+60^{\circ}+45^{\circ}$

Answer 3

Use the simetry of arcs in the trigonometric circle and the reduction to first quadrant. Finally, the formules of operations of arcs. For example: $$\sin 285^0=-\sin(360^0-285^0)=-\sin 75^0=-\sin(30^0+45^0)=$$ $$=-[\sin 30^0\cos 45^0+\sin 45^0\cos 30^0]=...$$

Note that the first equality is because the arc of $285^0$ is of $4^{\underline{\text{o}}}$ quadrant and the sine is negative and so the correspondent arc in the first quadrant is $360^0-285^0=75^0=30^0+45^0$

and for $\cos 285^0$: $$\cos 285^0=+\cos (360^0-285^0)=+\cos 75^0=+\cos (30^0+45^0)= $$ $$=\cos 30^0\cos 45^0-\sin 30^0\sin 45^0=... $$

etc.