# If \[\tan \theta = \dfrac{{24}}{7}\], then find the value of \[\sin \theta + \cos \theta \].

Answer

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Hint: - Use Pythagoras theorem, $\left[ {{{\left( {{\text{Perpendicular}}} \right)}^2} + {{\left( {{\text{Base}}} \right)}^2} = {{\left( {{\text{Hypotenuse}}} \right)}^2}} \right]$

Let, ABC is a right angle triangle, in which $\angle {\text{C}} = {90^0}$, and $\angle {\text{B = }}\theta $

Given:

\[\tan \theta = \dfrac{{24}}{7}\]

Therefore in $\Delta {\text{ABC, }}\tan \theta = \dfrac{{{\text{AC}}}}{{{\text{BC}}}}$

$ \Rightarrow {\text{AC = 24, and BC = 7}}$

Apply Pythagoras Theorem in $\Delta {\text{ABC}}$

$\left[ {{{\left( {{\text{Perpendicular}}} \right)}^2} + {{\left( {{\text{Base}}} \right)}^2} = {{\left( {{\text{Hypotenuse}}} \right)}^2}} \right]$

$

\Rightarrow {\text{A}}{{\text{B}}^2} = {\text{A}}{{\text{C}}^2} + {\text{B}}{{\text{C}}^2} \\

\Rightarrow {\text{A}}{{\text{B}}^2} = {24^2} + {7^2} = 625 = {\left( {25} \right)^2} \\

\Rightarrow {\text{AB = 25}} \\

$

Therefore in $\Delta {\text{ABC}}$, $\sin \theta = \dfrac{{{\text{AC}}}}{{{\text{AB}}}}$ and $\cos \theta = \dfrac{{{\text{BC}}}}{{{\text{AB}}}}$

So, the value of $\sin \theta + {\text{ }}\cos \theta = \dfrac{{24}}{{25}} + \dfrac{7}{{25}} = \dfrac{{31}}{{25}}$

So, this is the required answer.

Note- In such types of questions always apply Pythagoras Theorem which is stated above, then calculate the value of $\sin \theta $ which is perpendicular divided by hypotenuse, then calculate the value of $\cos \theta $ which is base divided by hypotenuse, then add these values we will get the required answer.

Let, ABC is a right angle triangle, in which $\angle {\text{C}} = {90^0}$, and $\angle {\text{B = }}\theta $

Given:

\[\tan \theta = \dfrac{{24}}{7}\]

Therefore in $\Delta {\text{ABC, }}\tan \theta = \dfrac{{{\text{AC}}}}{{{\text{BC}}}}$

$ \Rightarrow {\text{AC = 24, and BC = 7}}$

Apply Pythagoras Theorem in $\Delta {\text{ABC}}$

$\left[ {{{\left( {{\text{Perpendicular}}} \right)}^2} + {{\left( {{\text{Base}}} \right)}^2} = {{\left( {{\text{Hypotenuse}}} \right)}^2}} \right]$

$

\Rightarrow {\text{A}}{{\text{B}}^2} = {\text{A}}{{\text{C}}^2} + {\text{B}}{{\text{C}}^2} \\

\Rightarrow {\text{A}}{{\text{B}}^2} = {24^2} + {7^2} = 625 = {\left( {25} \right)^2} \\

\Rightarrow {\text{AB = 25}} \\

$

Therefore in $\Delta {\text{ABC}}$, $\sin \theta = \dfrac{{{\text{AC}}}}{{{\text{AB}}}}$ and $\cos \theta = \dfrac{{{\text{BC}}}}{{{\text{AB}}}}$

So, the value of $\sin \theta + {\text{ }}\cos \theta = \dfrac{{24}}{{25}} + \dfrac{7}{{25}} = \dfrac{{31}}{{25}}$

So, this is the required answer.

Note- In such types of questions always apply Pythagoras Theorem which is stated above, then calculate the value of $\sin \theta $ which is perpendicular divided by hypotenuse, then calculate the value of $\cos \theta $ which is base divided by hypotenuse, then add these values we will get the required answer.

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