No calculators allowed, especially graphing calculators

Geometry Level 2

Suppose that sin ⁡ α = 4 5 \sin \alpha = \dfrac 45 and sin ⁡ β = 1 2 \sin \beta = \dfrac 12 , where π 2 < α < β < π \dfrac \pi 2 < \alpha < \beta< \pi , find sin ⁡ ( α − β ) \sin (\alpha - \beta) .

− 4 3 + 3 10 -\frac { 4\sqrt { 3 } +3 }{ 10 } − 4 3 + 3 10 \frac { -4\sqrt { 3 } +3 }{ 10 } 4 3 − 3 10 \frac { 4\sqrt { 3 } -3 }{ 10 } 4 3 + 3 10 \frac { 4\sqrt { 3 } +3 }{ 10 }

César Castro by César Castro , 20, USA

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Apr 18, 2018

Let sin ⁡ a = 4 5 \sin a = \dfrac 45 and sin ⁡ b = 1 2 \sin b = \dfrac 12 , where 0 < b < a < π 2 0 < b < a < \dfrac \pi 2 . Then α = π − a \alpha = \pi - a and β = π − b \beta = \pi -b , and

sin ⁡ ( α − β ) = sin ⁡ ( π − a − π + b ) = sin ⁡ ( b − a ) = sin ⁡ b cos ⁡ a − cos ⁡ b sin ⁡ a = 1 2 × 3 5 − 3 2 × 4 5 = − 4 3 + 3 10 \begin{aligned} \sin (\alpha - \beta) & = \sin (\pi - a - \pi + b) \\ & = \sin (b-a) \\ & = \sin b \cos a - \cos b \sin a \\ & = \frac 12 \times \frac 35 - \frac {\sqrt 3}2 \times \frac 45 \\ & = \boxed{\dfrac {-4\sqrt 3+3}{10}} \end{aligned}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...