If \[\tan \theta = \dfrac{{24}}{7}\], then find the value of \[\sin \theta + \cos \theta \].
Answer
384k+ views
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.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

What is pollution? How many types of pollution? Define it

Give 10 examples for herbs , shrubs , climbers , creepers

Which state has the longest coastline in India A Tamil class 10 social science CBSE

Write an application to the principal requesting five class 10 english CBSE
