Answer
Verified
421.8k+ views
Hint: We use the trigonometric identity of \[\sin A\cos B + \cos A\sin B\] to simplify the given trigonometric equation. Compare the given equation with general trigonometric identity and write the value of A and B. Add the value of angles inside the bracket after obtaining the value from identity.
* If A and B are two angles then \[\sin A\cos B + \cos A\sin B = \sin (A + B)\]
Complete step-by-step answer:
We have to simplify the value of \[\cos {35^ \circ }\sin {55^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]
We can shuffle the terms and write \[\sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]
Since we can see that the given equation matches with the trigonometric identity \[\sin A\cos B + \cos A\sin B\], then on comparing we get the value of \[A = {55^ \circ };B = {35^ \circ }\]
Also, we know that \[\sin A\cos B + \cos A\sin B = \sin (A + B)\]
Substituting the value of A and B in right hand side of the identity
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = \sin ({55^ \circ } + {35^ \circ })\]
Add the value of angles inside the bracket in right hand side of the equation
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = \sin {90^ \circ }\]
Now substitute the value of \[\sin {90^ \circ } = 1\] in the right hand side of the equation.
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = 1\]
\[\therefore \]The value of \[\cos {35^ \circ }\sin {55^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]on simplification is equal to 1.
Note:
Many students make the mistake of calculating the value of cosine of the angle given and sine of the angle given using a scientific calculator and then substitute in the equation, then they multiply and add the terms. Keep in mind this is not the appropriate method as the values will be in decimal form and multiplying decimal values with decimal values will again give a long solution, students are advised to avoid this long calculation and make use of the trigonometric identity.
Also, many students who don’t remember the value of sine of angle obtained at the end can take help of the table that gives values of some common trigonometric functions at a few angles.
* If A and B are two angles then \[\sin A\cos B + \cos A\sin B = \sin (A + B)\]
Complete step-by-step answer:
We have to simplify the value of \[\cos {35^ \circ }\sin {55^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]
We can shuffle the terms and write \[\sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]
Since we can see that the given equation matches with the trigonometric identity \[\sin A\cos B + \cos A\sin B\], then on comparing we get the value of \[A = {55^ \circ };B = {35^ \circ }\]
Also, we know that \[\sin A\cos B + \cos A\sin B = \sin (A + B)\]
Substituting the value of A and B in right hand side of the identity
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = \sin ({55^ \circ } + {35^ \circ })\]
Add the value of angles inside the bracket in right hand side of the equation
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = \sin {90^ \circ }\]
Now substitute the value of \[\sin {90^ \circ } = 1\] in the right hand side of the equation.
\[ \Rightarrow \sin {55^ \circ }\cos {35^ \circ } + \cos {55^ \circ }\sin {35^ \circ } = 1\]
\[\therefore \]The value of \[\cos {35^ \circ }\sin {55^ \circ } + \cos {55^ \circ }\sin {35^ \circ }\]on simplification is equal to 1.
Note:
Many students make the mistake of calculating the value of cosine of the angle given and sine of the angle given using a scientific calculator and then substitute in the equation, then they multiply and add the terms. Keep in mind this is not the appropriate method as the values will be in decimal form and multiplying decimal values with decimal values will again give a long solution, students are advised to avoid this long calculation and make use of the trigonometric identity.
Also, many students who don’t remember the value of sine of angle obtained at the end can take help of the table that gives values of some common trigonometric functions at a few angles.
Angles (in degrees) | ${0^ \circ }$ | ${30^ \circ }$ | ${45^ \circ }$ | ${60^ \circ }$ | ${90^ \circ }$ |
sin | 0 | $\dfrac{1}{2}$ | $\dfrac{1}{{\sqrt 2 }}$ | $\dfrac{{\sqrt 3 }}{2}$ | $1$ |
cos | 1 | $\dfrac{{\sqrt 3 }}{2}$ | $\dfrac{1}{{\sqrt 2 }}$ | $\dfrac{1}{2}$ | 0 |
tan | 0 | $\dfrac{1}{{\sqrt 3 }}$ | 1 | $\sqrt 3 $ | Not defined |
cosec | Not defined | 2 | \[\sqrt 2 \] | \[\dfrac{2}{{\sqrt 3 }}\] | 1 |
sec | 1 | \[\dfrac{2}{{\sqrt 3 }}\] | \[\sqrt 2 \] | 2 | Not defined |
cot | Not defined | $\sqrt 3 $ | 1 | \[\dfrac{1}{{\sqrt 3 }}\] | 0 |
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE