Question

The smallest value of \[5\cos \theta +12\] is
A. \[5\]
B. \[12\]
C. \[7\]
D. \[17\]

Answer 1

Hint: Firstly we will find out the range in which \[\cos \theta \] lies that is \[-1\] to \[+1\] then multiply both side with \[5\] and then add \[12\] on both side of the term after that check out which is the smallest value obtained and then check which options is correct in the above given four options.

Complete step by step answer:
Trigonometry is that branch of mathematics which deals with the measurement of angles and the problems allied with angles. In the world of mathematics the word trigonometry is derived from the greek word. The earliest known work on trigonometry was recorded in Egypt. Trigonometry is used in astronomy and surveying, in finding height of the objects and in various branches of engineering. Trigonometry is also used to measure the distance that could not be measured directly. Trigonometry is the combination of trigono that is a triangle and metry that is measured.
Aryabhata the famous mathematician discovered sine and cosine.
Trigonometric ratios are the angles of a triangle; they relate the angles of a triangle to the lengths of its sides. Trigonometric ratios are important in the study of triangles and modeling periodic phenomena among many other applications.
The equation involving trigonometric ratios of an angle is called trigonometric identity if it is true for all values of the angle.
We know that there are a total six parts of a triangle out of this there are three angles and three sides. These are called parts or elements of a triangle.
Now according to the question :
We have given that we have to find the smallest value of \[5\cos \theta +12\]
As we know that the range of \[\cos \theta \] lies between \[-1\] to \[+1\] therefore:
\[\Rightarrow -1\le \cos \theta \le 1\]
Now multiply the whole term by \[5\] :
\[\Rightarrow -5\le 5\cos \theta \le 5\]
Now add \[12\] in the whole term:
\[\Rightarrow -5+12\le 5\cos \theta +12\le 5+12\]
\[\Rightarrow 7\le 5\cos \theta +12\le 17\]
Here you can see that the minimum value is \[7\] and maximum value is \[17\]

So, the correct answer is “Option C”.

Note:
We must keep one thing in mind that \[\sin \theta \] is not the same as \[\sin \times \theta \] because it represents a ratio, not a product and this is true for all the trigonometric ratios. Any trigonometric function of angle \[{{\theta }^{\circ }}\] is equal to the same trigonometric function of any angle \[n\times {{360}^{\circ }}+\theta \], where \[n\] is any integer.