Notes: `sin(135 - 30), sin(135) - sin(30)` Find the exact value of the expression.

Thursday, 24 September 2015

$\displaystyle{\sin{{\left({135}-{30}\right)}}},{\sin{{\left({135}\right)}}}-{\sin{{\left({30}\right)}}}$ Find the exact value of the expression.

$\displaystyle{\sin{{\left({{135}}^{{0}}-{{30}}^{{0}}\right)}}}={{\sin{{135}}}}^{{0}}\cdot{{\cos{{30}}}}^{{0}}-{{\cos{{135}}}}^{{0}}\cdot{{\sin{{30}}}}^{{0}}$

(we know that

$\displaystyle={\sin{{\left({90}+{45}\right)}}}\cdot{\cos{{30}}}-{\cos{{\left({90}+{45}\right)}}}\cdot{{\sin{{30}}}}^{{0}}$

$\displaystyle={{\cos{{45}}}}^{{0}}\cdot{\cos{{30}}}-{\left(-{{\sin{{45}}}}^{{0}}\right)}\cdot{{\sin{{30}}}}^{{0}}$

$\displaystyle=\frac{{1}}{\sqrt{{2}}}\cdot\frac{\sqrt{{3}}}{{2}}+\frac{{1}}{\sqrt{{2}}}\cdot\frac{{1}}{{2}}\ldots$

$\displaystyle{\sin{{\left({{135}}^{{0}}-{{30}}^{{0}}\right)}}}={{\sin{{135}}}}^{{0}}\cdot{{\cos{{30}}}}^{{0}}-{{\cos{{135}}}}^{{0}}\cdot{{\sin{{30}}}}^{{0}}$

(we know that

$\displaystyle={\sin{{\left({90}+{45}\right)}}}\cdot{\cos{{30}}}-{\cos{{\left({90}+{45}\right)}}}\cdot{{\sin{{30}}}}^{{0}}$

$\displaystyle={{\cos{{45}}}}^{{0}}\cdot{\cos{{30}}}-{\left(-{{\sin{{45}}}}^{{0}}\right)}\cdot{{\sin{{30}}}}^{{0}}$

$\displaystyle=\frac{{1}}{\sqrt{{2}}}\cdot\frac{\sqrt{{3}}}{{2}}+\frac{{1}}{\sqrt{{2}}}\cdot\frac{{1}}{{2}}$

$\displaystyle=\frac{{\sqrt{{3}}+{1}}}{{{2}\sqrt{{2}}}}={0.9661}$

lets check with direct values. $\displaystyle{\sin{{\left({{105}}^{{0}}\right)}}}={0.9659}\approx{0.966}$

Next expression is

$\displaystyle{\sin{{\left({{135}}^{{0}}\right)}}}-{\sin{{\left({{30}}^{{0}}\right)}}}$

$\displaystyle{\sin{{\left({{135}}^{{0}}\right)}}}-{\sin{{\left({{30}}^{{0}}\right)}}}={\sin{{\left({{90}}^{{0}}+{{45}}^{{0}}\right)}}}-{{\sin{{30}}}}^{{0}}$