Question

If \[sin{\text{ }}\theta {\text{ }} = {\text{ }}\dfrac{{12}}{{13}}{\text{ }},{\text{ }}\left( {{\text{ }}0{\text{ }} < {\text{ }}\theta {\text{ }} < {\text{ }}\dfrac{\pi }{2}} \right)\] and \[cos{\text{ }}\phi {\text{ }} = {\text{ }}\dfrac{{ - 3}}{5}{\text{ }},{\text{ }}\left( {{\text{ }}\pi {\text{ }} < {\text{ }}\phi {\text{ }} < {\text{ }}\dfrac{{3\pi }}{2}{\text{ }}} \right)\] , then \[sin{\text{ }}\left( {{\text{ }}\phi + {\text{ }}\theta {\text{ }}} \right)\] will be
 \[\left( 1 \right)\] \[\dfrac{{ - 56}}{{61}}\]
 \[\;\left( 2 \right)\] \[\dfrac{{ - 56}}{{65}}\]
 \[\;\left( 3 \right)\] \[\dfrac{1}{{65}}\]
 \[\left( 4 \right)\] \[ - 56\]

Answer 1

Hint: We have to find the value of \[sin{\text{ }}\left( {{\text{ }}\phi {\text{ }} + {\text{ }}\theta {\text{ }}} \right)\] . We solve this using the concept of the quadrant system . We should know the concept of sign and value of the trigonometric functions in four quadrants . The values of the trigonometric function have different values for different trigonometric functions with different signs .
We also apply the formula of \[sin{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right)\] and putting the values in the required formula we get the value .

Complete step-by-step answer:
 \[sin{\text{ }}\theta {\text{ }} = \dfrac{{12}}{{13}}{\text{ }},{\text{ }}\left( {{\text{ }}0{\text{ }} < {\text{ }}\theta {\text{ }} < {\text{ }}\dfrac{\pi }{2}{\text{ }}} \right)\]
 \[cos{\text{ }}\phi {\text{ }} = {\text{ }}\dfrac{{ - 3}}{5},{\text{ }}\left( {{\text{ }}\pi {\text{ }} < {\text{ }}\phi < {\text{ }}\dfrac{{3\pi }}{2}} \right)\]
The value of $ \theta $ lies in the first quadrant
We also know that ,
 \[sin{\text{ }}\theta {\text{ }} = {\text{ }}\;\dfrac{{{\text{ }}perpendicular}}{{hypotenuse\;}}\]
Comparing the two
Perpendicular \[\left( {{\text{ }}P{\text{ }}} \right){\text{ }} = {\text{ }}12\] and hypotenuse \[\left( {{\text{ }}H{\text{ }}} \right){\text{ }} = {\text{ }}13\]

Using the formula ,
 $ {(base)^2} + {(P)^2} = {(H)^2} $
So ,
Value of base \[\left( {{\text{ }}B{\text{ }}} \right)\] $ = \sqrt {[{H^2} - {P^2}] } $
 $ B = \sqrt {[{{13}^2} - {{12}^2}] } $
 $ B = \sqrt {[169 - 144] } $
 $ B = \sqrt {[25] } $
 \[B{\text{ }} = {\text{ }}5\]
As the value of cos is positive in first quadrant , then
 \[cos{\text{ }}\theta {\text{ }} = {\text{ }}\dfrac{B}{H}\]
 \[cos{\text{ }}\theta {\text{ }} = {\text{ }}\dfrac{5}{{13}}\]
Similarly , calculating the value of \[sin{\text{ }}\phi \]

As ,
 \[cos{\text{ }}\phi = {\text{ }}\dfrac{{ - 3}}{5}\]
As $ \phi $ lies in third quadrant
We also know that ,
 \[cos{\text{ }}\phi {\text{ }} = \dfrac{B}{H}\]
Comparing the two
Base \[ = {\text{ }}3\] and hypotenuse \[ = {\text{ }}5\]

Using the formula of hypotenuse
 $ {B^2} + {P^2} = {H^2} $
So ,
Value of $ P = \sqrt {[{5^2} - {3^2}] } $
 $ P = \sqrt {[25 - 9] } $
 $ P = \sqrt {[16] } $
 \[P{\text{ }} = {\text{ }}4\]
As the value of sin is negative in third quadrant , then
 \[sin{\text{ }}\phi {\text{ }} = {\text{ }}\dfrac{P}{H}\]
 \[sin{\text{ }}\phi {\text{ }} = {\text{ }}\dfrac{{ - 4}}{5}\]
Using the formula
 \[sin{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right){\text{ }} = {\text{ }}sin{\text{ }}a{\text{ }} \times {\text{ }}cos{\text{ }}b{\text{ }} + {\text{ }}sin{\text{ }}b{\text{ }} \times {\text{ }}cos{\text{ }}a\]

Now putting the values in the formula , we get
 \[sin{\text{ }}\left( {{\text{ }}\theta {\text{ }} + {\text{ }}\phi } \right){\text{ }} = {\text{ }}sin{\text{ }}\theta {\text{ }} \times cos{\text{ }}\phi {\text{ }} + {\text{ }}sin{\text{ }}\phi {\text{ }} \times cos{\text{ }}\theta \]
Substituting the values in the formula , we get
 \[sin{\text{ }}\left( {{\text{ }}\theta {\text{ }} + {\text{ }}\phi {\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{12}}{{13}}{\text{ }} \times {\text{ }}\left( {{\text{ }}\dfrac{{ - 3}}{5}} \right){\text{ }} + {\text{ }}\left( {{\text{ }}\dfrac{{ - 4}}{5}{\text{ }}} \right){\text{ }} \times {\text{ }}\dfrac{5}{{13}}\]
 \[sin{\text{ }}\left( {{\text{ }}\theta {\text{ }} + {\text{ }}\phi {\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{ - 36}}{{65}}{\text{ }} - \dfrac{{20}}{{65}}\]
 \[sin{\text{ }}\left( {{\text{ }}\theta {\text{ }} + {\text{ }}\phi {\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{ - 56}}{{65}}\]
Hence , the value of \[sin{\text{ }}\left( {{\text{ }}\theta {\text{ }} + {\text{ }}\phi {\text{ }}} \right){\text{ }} = {\text{ }}\dfrac{{ - 56}}{{65}}\]
Thus , the correct option is \[\left( 2 \right)\]
So, the correct answer is “Option 2”.

Note: We have various trigonometric formulas used to solve the problem
The various trigonometric formulas used :
 \[sin{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right){\text{ }} = {\text{ }}sin{\text{ }}a{\text{ }} \times {\text{ }}cos{\text{ }}b{\text{ }} + {\text{ }}sin{\text{ }}b{\text{ }} \times {\text{ }}cos{\text{ }}a\]
 \[sin{\text{ }}\left( {{\text{ }}a{\text{ }} - {\text{ }}b{\text{ }}} \right){\text{ }} = {\text{ }}sin{\text{ }}a{\text{ }} \times {\text{ }}cos{\text{ }}b{\text{ }} - {\text{ }}sin{\text{ }}b{\text{ }} \times {\text{ }}cos{\text{ }}a\]
 \[cos{\text{ }}\left( {{\text{ }}a{\text{ }} + {\text{ }}b{\text{ }}} \right){\text{ }} = {\text{ }}cos{\text{ }}a{\text{ }} \times {\text{ }}cos{\text{ }}b{\text{ }} - {\text{ }}sin{\text{ }}b{\text{ }} \times {\text{ }}sin{\text{ }}a\]
 \[cos{\text{ }}\left( {{\text{ }}a{\text{ }} - {\text{ }}b{\text{ }}} \right){\text{ }} = {\text{ }}cos{\text{ }}a{\text{ }} \times {\text{ }}cos{\text{ }}b{\text{ }} + {\text{ }}sin{\text{ }}b{\text{ }} \times {\text{ }}sin{\text{ }}a\]
All the trigonometric functions are positive in first quadrant , the sin function are positive in second quadrant and rest are negative , the tan function are positive in third quadrant and rest are negative , the cos function are positive in fourth quadrant and rest are negative .