Answer
396.9k+ 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
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)