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Question
AnswersThe length of the shadow of a tree is $\sqrt {\text{3}} $ times its height, then find the angle of elevation of the sun at the same time.
Answer
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Hint: We will determine the ratio of length of shadow and height of tree and use the trigonometric ratio of ${\text{tan}\theta }$. As ${\text{tan}\theta }$ is the ratio of height of a triangle and it’s base, we will use it to represent height of tree and length of shadow in terms of tangent of angle of elevation. The value of angle of elevation can be calculated by the value of tangent of angle of elevation.
Complete step by step solution: Consider the following diagram for the situation mentioned in the question
In the figure, AB represents the tree, and AC represents the length of shadow casted.
Let, the height of tree be ${\text{h}}$ and length of shadow be ${\text{b}}$
So, ${\text{AB = h}}$and ${\text{AC = b}}$
Let, the angle of elevation of the sun be \[{\text{x}}\]
Then, in the figure, angle at C is ${\text{x}}$
Now, as given, the length of the shadow of a tree is $\sqrt 3 $ times its height,
So, ${\text{b = }}\sqrt {\text{3}} {\text{h}}$
Now, among various trigonometric ratios, those dealing with base and height are tangent and cotangent
We know that, for a right-angled triangle,
${\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}}$
Also, the triangle in the figure ABC is right angled at A
Base of the triangle is AC and height of the triangle is AB
So, applying ${\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}}$
Substituting, ${\theta = x}$
${\text{tanx = }}\dfrac{{{\text{AB}}}}{{{\text{AC}}}}$
${\text{tanx = }}\dfrac{{\text{h}}}{{\text{b}}}$
Now, substituting, ${\text{b = }}\sqrt {\text{3}} {\text{h}}$
${\text{tanx = }}\dfrac{{\text{h}}}{{\sqrt {\text{3}} {\text{h}}}}$
${\text{tanx = }}\dfrac{{\text{1}}}{{\sqrt {\text{3}} }}$
Now, we know that,
${\text{tan}}\left( {{\text{30}^\circ }} \right){\text{ = }}\dfrac{{\text{1}}}{{\sqrt {\text{3}} }}$
Therefore, ${\text{x = 30}^\circ }$
Hence, the angle of elevation of the sun is ${\text{30}^\circ}$
Note: While attempting the questions regarding trigonometric ratios, students must remember the definition of all of the trigonometric ratios. The following are the definition of all trigonometric ratios:
\[
{\text{sin}\theta = }\dfrac{{{\text{height}}}}{{{\text{hypotenuse}}}} \\
{\text{cos}\theta = }\dfrac{{{\text{base}}}}{{{\text{hypotenuse}}}} \\
{\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}} \\
{\text{sec}\theta = }\dfrac{{{\text{hypotenuse}}}}{{{\text{base}}}} \\
{\text{cosec}\theta = }\dfrac{{{\text{hypotenuse}}}}{{{\text{height}}}} \\
{\text{cot}\theta = }\dfrac{{{\text{base}}}}{{{\text{height}}}} \\
\]
Students must remember them to solve these types of questions.
Complete step by step solution: Consider the following diagram for the situation mentioned in the question
In the figure, AB represents the tree, and AC represents the length of shadow casted.
Let, the height of tree be ${\text{h}}$ and length of shadow be ${\text{b}}$
So, ${\text{AB = h}}$and ${\text{AC = b}}$
Let, the angle of elevation of the sun be \[{\text{x}}\]
Then, in the figure, angle at C is ${\text{x}}$
Now, as given, the length of the shadow of a tree is $\sqrt 3 $ times its height,
So, ${\text{b = }}\sqrt {\text{3}} {\text{h}}$
Now, among various trigonometric ratios, those dealing with base and height are tangent and cotangent
We know that, for a right-angled triangle,
${\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}}$
Also, the triangle in the figure ABC is right angled at A
Base of the triangle is AC and height of the triangle is AB
So, applying ${\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}}$
Substituting, ${\theta = x}$
${\text{tanx = }}\dfrac{{{\text{AB}}}}{{{\text{AC}}}}$
${\text{tanx = }}\dfrac{{\text{h}}}{{\text{b}}}$
Now, substituting, ${\text{b = }}\sqrt {\text{3}} {\text{h}}$
${\text{tanx = }}\dfrac{{\text{h}}}{{\sqrt {\text{3}} {\text{h}}}}$
${\text{tanx = }}\dfrac{{\text{1}}}{{\sqrt {\text{3}} }}$
Now, we know that,
${\text{tan}}\left( {{\text{30}^\circ }} \right){\text{ = }}\dfrac{{\text{1}}}{{\sqrt {\text{3}} }}$
Therefore, ${\text{x = 30}^\circ }$
Hence, the angle of elevation of the sun is ${\text{30}^\circ}$
Note: While attempting the questions regarding trigonometric ratios, students must remember the definition of all of the trigonometric ratios. The following are the definition of all trigonometric ratios:
\[
{\text{sin}\theta = }\dfrac{{{\text{height}}}}{{{\text{hypotenuse}}}} \\
{\text{cos}\theta = }\dfrac{{{\text{base}}}}{{{\text{hypotenuse}}}} \\
{\text{tan}\theta = }\dfrac{{{\text{height}}}}{{{\text{base}}}} \\
{\text{sec}\theta = }\dfrac{{{\text{hypotenuse}}}}{{{\text{base}}}} \\
{\text{cosec}\theta = }\dfrac{{{\text{hypotenuse}}}}{{{\text{height}}}} \\
{\text{cot}\theta = }\dfrac{{{\text{base}}}}{{{\text{height}}}} \\
\]
Students must remember them to solve these types of questions.