Question

How do you write the trigonometric form of 4?

Answer 1

Hint: Here, we will assume that the given number is a complex number. We will write the general equation of the trigonometric form of a complex number. Then, after substituting the given complex number 4 in the general equation, we will be able to find the required trigonometric form.

Formula Used:
$z = x + i \cdot y = r \cdot \left( {\cos \alpha + i \cdot \sin \alpha } \right)$

Complete step by step solution:
Except for 0, any complex number can be written in the trigonometric form or in polar coordinates:
$z = x + i \cdot y = r \cdot \left( {\cos \alpha + i \cdot \sin \alpha } \right)$
Here, $r = \sqrt {{x^2} + {y^2}} $
And, $\alpha = {\tan ^{ - 1}}\left( {\dfrac{y}{x}} \right)$
Now, we have to find the trigonometric form of 4.
Thus, we have $x = 4$ and $y = 0$
Therefore, substituting this in the above formula, we get,
$z = 4 + i\left( 0 \right) = 4$
Also,
$r = \sqrt {{4^2} + {0^2}} = \sqrt {16} = 4$
And,
$\alpha = {\tan ^{ - 1}}\left( {\dfrac{0}{4}} \right) = {\tan ^{ - 1}}\left( {\tan 0^\circ } \right) = 0^\circ $
Hence, in the trigonometric form, we get,
$z = 4 \cdot \left( {\cos 0^\circ + i\sin 0^\circ } \right)$

Hence, we write the trigonometric form of 4 as $z = 4 \cdot \left( {\cos 0^\circ + i\sin 0^\circ } \right)$
Thus, this is the required answer.

Additional information:
In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine and the cosine function. In simple terms they are written as ‘sin’, ‘cos’ and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.

Note:
A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit, satisfying the equation ${i^2} = - 1$. As we know, no real number can satisfy this equation, thus, $i$ is called an imaginary number also known as iota. Trigonometry is a branch of mathematics which helps us to study the relationship between the sides and the angles of a triangle.