
If \[A = B + C\] and the value of A, B, and C are 13, 12, and 5 respectively, then find the angle between A and C.
A.\[{\cos ^{ - 1}}\left( {\dfrac{5}{{13}}} \right)\]
B. \[{\cos ^{ - 1}}\left( {\dfrac{{13}}{{12}}} \right)\]
C. \[\dfrac{\pi }{2}\]
D. \[{\sin ^{ - 1}}\left( {\dfrac{5}{{12}}} \right)\]
Answer
164.4k+ views
Hints First we will apply the Pythagorean theorem to check whether the given triangle is a right-angle triangle or not. Then decide which are the legs and hypotenuse of the triangle. Then find the angle between B and C using the trigonometry ratios.
Formula used
The Pythagoras theorem for right angle is,
\[{a^2} + {b^2} = {c^2}\], where a is the base, b is the height, and c is the hypotenuse.
Also,
\[\cos \theta = \dfrac{p}{q}\], where p is the base and q is the hypotenuse.
Complete step by step solution
The given lengths of the sides are 13, 12, and 5.
Now,
\[{12^2} + {5^2}\]
\[ = 144 + 25\]
\[ = 169\]
\[ = {13^2}\]
Therefore, according to Pythagoras' theorem, the given triangle is right-angled.
The diagram of the given triangle is,

Use the formula \[\cos \theta = \dfrac{p}{q}\] , where p is the base and q is the hypotenuse to obtain the required result.
Therefore,
\[\cos \theta = \dfrac{5}{{13}}\]
\[\theta = {\cos ^{ - 1}}\left( {\dfrac{5}{{13}}} \right)\] .
The correct option is A.
Note Students often used cosine formula \[{A^2} = {B^2} + {C^2} + 2BC\cos \phi \] to obtain the angle between B and C and \[\theta\]. By using the cosine formula we cannot able find the angle between B and C. Because the cosine formula is applicable to an oblique triangle. Thus we will use trigonometry ratios to find the angle between them.
Formula used
The Pythagoras theorem for right angle is,
\[{a^2} + {b^2} = {c^2}\], where a is the base, b is the height, and c is the hypotenuse.
Also,
\[\cos \theta = \dfrac{p}{q}\], where p is the base and q is the hypotenuse.
Complete step by step solution
The given lengths of the sides are 13, 12, and 5.
Now,
\[{12^2} + {5^2}\]
\[ = 144 + 25\]
\[ = 169\]
\[ = {13^2}\]
Therefore, according to Pythagoras' theorem, the given triangle is right-angled.
The diagram of the given triangle is,

Use the formula \[\cos \theta = \dfrac{p}{q}\] , where p is the base and q is the hypotenuse to obtain the required result.
Therefore,
\[\cos \theta = \dfrac{5}{{13}}\]
\[\theta = {\cos ^{ - 1}}\left( {\dfrac{5}{{13}}} \right)\] .
The correct option is A.
Note Students often used cosine formula \[{A^2} = {B^2} + {C^2} + 2BC\cos \phi \] to obtain the angle between B and C and \[\theta\]. By using the cosine formula we cannot able find the angle between B and C. Because the cosine formula is applicable to an oblique triangle. Thus we will use trigonometry ratios to find the angle between them.
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