# Express \[{\text{sin}}{67^0}{\text{ }} + {\text{ cos}}{75^0}\] in terms of trigonometric ratios of angles

between \[{0^0}\] and \[{45^0}\].

Last updated date: 27th Mar 2023

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Answer

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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 \[\theta \in \left[ {0,{{90}^0}} \right].\]

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 \[\theta \in \left[ {0,{{90}^0}} \right].\]

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