Question

If we have a trigonometric expression as \[\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} = 3\], then \[\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3}\] is equal to

Answer 1

Hint: To solve this question first we assume a variable that is equal to the asked expression. For further solving we use the concept of the range of the trigonometric function from here we got that this relation exists if there the values of all trigonometry function is maximum at a particular value so from there we are able to find the values of all the angle then put all those values in the asked expression and we are able to find the value of \[\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3}\].

Complete step-by-step solution:
Given,
\[\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} = 3\]
To find,
The value of \[\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3}\]
The given equation is
\[\sin {\theta _1} + \sin {\theta _2} + \sin {\theta _3} = 3\]
We know that the range of sine functions is of \[ - 1\] and \[1\]. Example [\[ -1,1\]]
The sum of three sine functions is 3 that is given. So it is possible when all the sine functions have a maximum value that is 1.
Maximum value of a sine function is at \[{90^ \circ }\] and all the values of angles are \[{90^ \circ }\].
\[{\theta _1} = {\theta _2} = {\theta _3} = {90^ \circ }\]
Here, we get the values of all angles. Now on putting the values in the expression
Let \[x = \cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3}\] ……(i)
On putting all the values of angles in equation (i)
\[x = \cos {90^ \circ } + \cos {90^ \circ } + \cos {90^ \circ }\]
We know that the value of the cos function at \[{90^ \circ }\] is 0.
\[\cos {90^ \circ } = 1\] on putting this value in the obtained equation
\[x = 0 + 0 + 0\]
On further solving we get the value of the asked expression.
\[x = 0\]
Final answer:
According to the obtained answer the value of the given expression is
\[\cos {\theta _1} + \cos {\theta _2} + \cos {\theta _3} = 0\].

Note: This question is very tricky. We have to use the range and the maximum possible value of that function. This question does not strike directly in our mind. This is by the practice. Maximum students commit mistakes in these types of questions by taking the range and guessing the maximum value that is possible for that function.