Question

# Express ${\text{sin}}{67^0}{\text{ }} + {\text{ cos}}{75^0}$ in terms of trigonometric ratios of anglesbetween ${0^0}$ and ${45^0}$.

Hint:- Use ${\text{6}}{{\text{7}}^0} = {90^0} - {23^0}$and ${75^0} = {90^0} - {15^0}$.

We are given with the equation,
$\Rightarrow {\text{sin}}{67^0}{\text{ }} + {\text{ cos}}{75^0}$ ---->(1)
And, had to convert the angles of the given equation between ${0^0}$ and ${45^0}$.
We know that,
According to the trigonometric identities,
$\Rightarrow {\text{sin}}\left( {{{90}^0} - \theta } \right) = \cos \theta$ ---->(2)
$\Rightarrow$And, ${\text{cos}}\left( {{{90}^0} - \theta } \right) = \sin \theta$ ----->(3)
So, above equation 1 can be written as,
$\Rightarrow \sin {\left( {90 - 23} \right)^0} + \cos {\left( {90 - 15} \right)^0}$ ---->(4)
So, using equation 2 and 3. We can write equation 4 as,
$\Rightarrow {\text{cos}}{23^0} + {\text{sin1}}{5^0}$
Hence, the given equation is expressed in trigonometric ratios
of angles lying between ${0^0}$ and ${45^0}$.
$\Rightarrow$So, ${\text{sin}}{67^0}{\text{ }} + {\text{ cos}}{75^0} = {\text{cos}}{23^0} + {\text{sin1}}{5^0}$
Note:- Whenever we came up with this type of problem then remember that,
${\text{sin}}\left( {{{90}^0} - \theta } \right) = \cos \theta$ and ${\text{cos}}\left( {{{90}^0} - \theta } \right) = \sin \theta$. And value of${\text{ sin}}\theta ,{\text{ cos}}\theta ,{\text{ tan}}\theta ,$
${\text{cot}}\theta ,{\text{ cosec}}\theta$ and ${\text{sec}}\theta$ is always positive if $\theta \in \left[ {0,{{90}^0}} \right].$