Answer

Verified

455.1k+ views

**Hint:**Observe that\[{13^2} + {(3\sqrt 3 )^2} = {14^2}\].

Multiply and divide the given function by 14 to get a point on the unit circle. Also, use the fact: For every point $ P(x,y) $on the unit circle\[{x^2} + {y^2} = 1\]there exists $ \theta \in [0,2\pi ) $such that$ x = \cos \theta $ and $ y = \sin \theta $.

Simplify the function to get an expression without $ \sin x $using the trigonometric identity $ \cos (A - B) = \cos A\cos B - \sin A\sin B $. Finally, use the range of $ \cos \theta $ : $ - 1 \leqslant \cos (x - \theta ) \leqslant 1 $ and get the answer.

**Complete step by step solution:**

We are given a trigonometric function $ 13\cos x + 3\sqrt 3 \sin x - 4 $

We need to determine the range of this function.

Let $ f(x) = 13\cos x + 3\sqrt 3 \sin x - 4 $

A range of a function $ f $would be the set of all the outcomes or outputs for the various inputs in its domain. It is

Domain of a function $ f $is the set of all possible values on which $ f $can be applied.

This means if $ x $is a variable, then it is possible that for some values of $ x $, the function is not defined.

We would be simplifying the given function to ease the process of finding the range.

We can see that\[{13^2} + {(3\sqrt 3 )^2} = 169 + 27 = 196 = {14^2}\]

Therefore, we multiply and divide by 14 throughout the expression of the given function

Then, we get

$

f(x) = 13\cos x + 3\sqrt 3 \sin x - 4 \\

= 14(\dfrac{{13}}{{14}}\cos x + \dfrac{{3\sqrt 3 }}{{14}}\sin x) - 4..............(1) \\

$

Now, we can observe that $ {(\dfrac{{13}}{{14}})^2} + {(\dfrac{{3\sqrt 3 }}{{14}})^2} = \dfrac{{169}}{{196}} + \dfrac{{27}}{{196}} = \dfrac{{196}}{{196}} = 1 $

This implies that $ (\dfrac{{13}}{{14}},\dfrac{{3\sqrt 3 }}{{14}}) $is a point on the unit circle\[{x^2} + {y^2} = 1\]

Let us recall a fact here: For every point $ P(x,y) $on the unit circle\[{x^2} + {y^2} = 1\]there exists $ \theta \in [0,2\pi ) $ such that $ x = \cos \theta $ and $ y = \sin \theta $.

Therefore, for the point $ (\dfrac{{13}}{{14}},\dfrac{{3\sqrt 3 }}{{14}}) $, there exists $ \theta \in [0,2\pi ) $such that $ \dfrac{{13}}{{14}} = \cos \theta $ and $\dfrac{{3\sqrt 3 }}{{14}} = \sin \theta $.

Then, on substituting in equation (1), we get

$

f(x) = 14(\cos \theta \cos x + \sin \theta \sin x) - 4 \\

= 14(\cos x\cos \theta + \sin x\sin \theta ) - 4 \\

$

Here we have rearranged the cosine and sine values. We can do this because they are real numbers.

We will use the angle difference identity:

$ \cos (A - B) = \cos A\cos B - \sin A\sin B $

Then$ f(x) = 14(\cos (x - \theta )) - 4 $

Now, we know that the range of $ \cos \theta $ is $ [ - 1,1] $ for any angle $ \theta $

That is, $ - 1 \leqslant \cos \theta \leqslant 1 $ for any angle $ \theta $

Therefore,

$ - 1 \leqslant \cos (x - \theta ) \leqslant 1$

Multiplying 14 throughout the expression, we get

$ - 14 \leqslant 14\cos (x - \theta ) \leqslant 14$

Subtracting 4 from each value, we get

$

- 14 + 4 \leqslant 14\cos (x - \theta ) + 4 \leqslant 14 + 4 \\

\Rightarrow - 18 \leqslant 14\cos (x - \theta ) + 4 \leqslant 18 \\

$

Hence, the range of $ f(x) = 13\cos x + 3\sqrt 3 \sin x - 4$ is $ [ - 18,18] $.

**Note:**The inequality in the final step means that every real number between -18 and 18 belongs to the range of $ f(x) = 13\cos x + 3\sqrt 3 \sin x - 4 $. Therefore, writing {-18, 18} as an answer is completely wrong.

Here, $ [ - 18,18] $ indicates the closed interval taking every value from -18 to 18.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

The states of India which do not have an International class 10 social science 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

One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

Name the three parallel ranges of the Himalayas Describe class 9 social science CBSE