Answer
Verified
377.7k+ views
Hint: Here, this problem is solved with the trigonometric formulas. Firstly, we consider the above equation is equal to arbitrary constant ‘t’, then equalizing all, and using this trigonometric formula \[\cos C+\cos D=2\cos \left( \dfrac{C+D}{2} \right)\cos \left( \dfrac{C-D}{2} \right)\] , then we evaluate the \[x+y+z\]and finally we find the correct option.
Complete step by step answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric functions.
Let us assume the given expression is equal to arbitrary constant ’t’
\[\dfrac{x}{\cos \alpha }=\dfrac{y}{\cos \left( \alpha -\dfrac{2\pi }{3} \right)}=\dfrac{z}{\cos \left( \alpha +\dfrac{2\pi }{3} \right)}=t\]
Here, we are equalizing all the individual terms to arbitrary constant ’t’, we are multiplied the denominator part of the x, y and z terms with constants ’t’, then we get
\[x=t\cos \alpha \]
\[y=t\cos \left( \alpha -\dfrac{2\pi }{3} \right)\]
\[z=t\cos \left( \alpha +\dfrac{2\pi }{3} \right)\]
Now, we are solving for the expression \[x+y+z\], substituting the x, y and z values in the \[x+y+z\]
\[\Rightarrow x+y+z=t\cos \alpha +t\cos \left( \alpha -\dfrac{2\pi }{3} \right)+t\cos \left( \alpha +\dfrac{2\pi }{3} \right)\]
Here, we are taking the ‘t’ common from Right Hand Side, then we obtain
\[\Rightarrow x+y+z=t\left( \cos \alpha +\cos \left( \alpha -\dfrac{2\pi }{3} \right)+\cos \left( \alpha +\dfrac{2\pi }{3} \right) \right)\]
From the above equation, we have to apply the basic formula to evaluate,
As we now the cosine sum identity i.e., \[\cos C+\cos D=2\cos \left( \dfrac{C+D}{2} \right)\cos \left( \dfrac{C-D}{2} \right)\]
Evaluating the expression, on basis of the cosine sum identity We get
Here, \[C=\alpha \]and \[D=\left( \dfrac{2\pi }{3} \right)\]
Substituting the Cand D values in the expression we get
\[\Rightarrow t\left( \cos \alpha +2\cos \alpha \cos \left( \dfrac{-2\pi }{3} \right) \right)\]
We know that, \[\cos \left( \dfrac{-2\pi }{3} \right)=\dfrac{-1}{2}\]
On substituting,
\[\Rightarrow t\left( \cos \alpha +2\cos \alpha \left( \dfrac{-1}{2} \right) \right)\]
\[\Rightarrow t\left( \cos \alpha -\cos \alpha \right)\]
\[=0\]
Therefore, \[x+y+z=0\]
So, the correct answer is “Option B”.
Note: When solving trigonometry-based questions, we have to know the definitions of ratios and always remember the standard angles and formulas are useful for solving certain integration problems where a cosine and sum identity may make things much simpler to solve. Thus, in math as well as physics, these formulae are useful to derive many important identities.
Complete step by step answer:
A function of an angle expressed as the ratio of two of the sides of a right triangle that contains that angle; the sine, cosine, tangent, cotangent, secant, or cosecant known as trigonometric functions.
Let us assume the given expression is equal to arbitrary constant ’t’
\[\dfrac{x}{\cos \alpha }=\dfrac{y}{\cos \left( \alpha -\dfrac{2\pi }{3} \right)}=\dfrac{z}{\cos \left( \alpha +\dfrac{2\pi }{3} \right)}=t\]
Here, we are equalizing all the individual terms to arbitrary constant ’t’, we are multiplied the denominator part of the x, y and z terms with constants ’t’, then we get
\[x=t\cos \alpha \]
\[y=t\cos \left( \alpha -\dfrac{2\pi }{3} \right)\]
\[z=t\cos \left( \alpha +\dfrac{2\pi }{3} \right)\]
Now, we are solving for the expression \[x+y+z\], substituting the x, y and z values in the \[x+y+z\]
\[\Rightarrow x+y+z=t\cos \alpha +t\cos \left( \alpha -\dfrac{2\pi }{3} \right)+t\cos \left( \alpha +\dfrac{2\pi }{3} \right)\]
Here, we are taking the ‘t’ common from Right Hand Side, then we obtain
\[\Rightarrow x+y+z=t\left( \cos \alpha +\cos \left( \alpha -\dfrac{2\pi }{3} \right)+\cos \left( \alpha +\dfrac{2\pi }{3} \right) \right)\]
From the above equation, we have to apply the basic formula to evaluate,
As we now the cosine sum identity i.e., \[\cos C+\cos D=2\cos \left( \dfrac{C+D}{2} \right)\cos \left( \dfrac{C-D}{2} \right)\]
Evaluating the expression, on basis of the cosine sum identity We get
Here, \[C=\alpha \]and \[D=\left( \dfrac{2\pi }{3} \right)\]
Substituting the Cand D values in the expression we get
\[\Rightarrow t\left( \cos \alpha +2\cos \alpha \cos \left( \dfrac{-2\pi }{3} \right) \right)\]
We know that, \[\cos \left( \dfrac{-2\pi }{3} \right)=\dfrac{-1}{2}\]
On substituting,
\[\Rightarrow t\left( \cos \alpha +2\cos \alpha \left( \dfrac{-1}{2} \right) \right)\]
\[\Rightarrow t\left( \cos \alpha -\cos \alpha \right)\]
\[=0\]
Therefore, \[x+y+z=0\]
So, the correct answer is “Option B”.
Note: When solving trigonometry-based questions, we have to know the definitions of ratios and always remember the standard angles and formulas are useful for solving certain integration problems where a cosine and sum identity may make things much simpler to solve. Thus, in math as well as physics, these formulae are useful to derive many important identities.
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
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE