Question

Evaluate : cos 105* + sin 105*

Answer 1

sin(60+45) + cos(60+45)

=[sin60cos45 +cos60sin45] + [cos60 cos45 - sin60sin45]

=cos 60 sin 45 +cos60 cos45

=1/2   *  1/sqrt2   +  1/2  * 1/sqrt2

=1/sqrt2

cos45

Answer 2

The value of $\cos 105^{\circ}+\sin 105^{\circ} \text { is } \bold{\frac{1}{\sqrt{2}}}$

Cos trigonometric values are positive in “first and fourth” quadrants in coordinate axes.

Sin trigonometric values are positive in “first and second” quadrants in coordinate axes.

$\cos 105^{\circ}+\sin 105^{\circ}=\cos \left(90^{\circ}+15^{\circ}\right)+\sin \left(90^{\circ}+15^{\circ}\right)$

$=-\sin \left(15^{\circ}\right)+\cos \left(15^{\circ}\right)\quad \left(a s \cos \left(90^{\circ}+\theta\right)=-\sin \theta \text { and } \sin \left(90^{\circ}+\theta\right)=\cos \theta\right)$

$=\cos 15^{\circ}-\sin 15^{\circ}$

$=\sqrt{2}\left[\frac{1}{\sqrt{2}} \cos 15^{\circ}-\frac{1}{\sqrt{2}} \sin 15^{\circ}\right]\quad (Multiply\ and\ divide\ by\ \sqrt{2})$

$=\sqrt{2}\left[\sin 45^{\circ} \cos 15^{\circ}-\cos 45^{\circ} \sin 15^{\circ}\right]\quad\left({As} \sin 45^{\circ}=\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right)$

$=\sqrt{2}\left[\sin \left(45^{\circ}-15^{\circ}\right)\right]\quad ({As} \sin (A-B)=\sin A \cos B-\cos A \sin B)$

$=\sqrt{2}\left[\sin 30^{\circ}\right]$

$=\sqrt{2} \times \frac{1}{2}=\frac{1}{\sqrt{2}}$