Question

if cos alpha=8/17 and sin beta=5/13 then find the value of sin (alpha-beta)​

Answer 1

Given:-

• $\bf cos\alpha=\dfrac {8}{17}$
• $\bf sin\beta=\dfrac {5}{13}$

To find:-

• $\bf sin (\alpha-\beta)$

Solution:-

We know that

• $\boxed{\bf sin\alpha=\sqrt {1-cos^2\alpha}}$

Substitute the values

$\\\qquad\quad\displaystyle\sf{:}\implies sin\alpha=\sqrt {1-(\dfrac {8}{17})^2 }$

$\\\qquad\quad\displaystyle\sf{:}\implies sin\alpha=\sqrt {1-\dfrac{64}{289}}$

$\\\qquad\quad\displaystyle\sf{:}\implies sin\alpha=\sqrt {\dfrac {289-64}{289}}$

$\\\qquad\quad\displaystyle\sf{:}\implies sin\alpha=\sqrt {\dfrac{225}{289}}$

$\\\qquad\quad\displaystyle\sf{:}\implies sin\alpha=\dfrac {15}{17}$

Again

$\boxed{\bf cos\beta=\sqrt {1-sin^2\beta}}$

• Substitute the values

$\\\qquad\quad\displaystyle\sf{:}\implies cos\beta=\sqrt {1-(\dfrac {5}{13})^2}$

$\\\qquad\quad\displaystyle\sf{:}\implies cos\beta=\sqrt {1-\dfrac {25}{289}}$

$\\\qquad\quad\displaystyle\sf{:}\implies cos\beta=\sqrt {\dfrac {1-169-25}{169}}$

$\\\qquad\quad\displaystyle\sf{:}\implies cos\beta=\sqrt {\dfrac {144}{169}}$

$\\\qquad\quad\displaystyle\sf{:}\implies cos\beta=\dfrac {13}{19}$

• Now

$\boxed {\bf sin (\alpha-\beta)=sin\alpha.cos\beta-cos\alpha.sin\beta}$

• Substitute the values

$\\\qquad\quad\displaystyle\sf{:}\implies sin (\alpha-\beta)=\dfrac {15}{17}\times\dfrac {12}{13}-\dfrac {8}{17}\times\dfrac {5}{13}$

$\\\qquad\quad\displaystyle\sf{:}\implies sin (\alpha-\beta)=\dfrac {180}{221}-\dfrac {40}{221}$

$\\\qquad\quad\displaystyle\sf{:}\implies sin (\alpha-\beta)=\dfrac{180-40}{221}$

$\\\qquad\quad\displaystyle\bf{:}\implies sin (\alpha-\beta)=\dfrac{140}{221}$