Question

How do you find the $6$ trigonometric functions for $- 45$ degrees?

Answer 1

Hint : Trigonometric functions are the real functions which are related to an angle of a right angled triangle with the ratios of the two side lengths. Six trigonometric functions include sine, cosine, tangent, cot, secant and cosec functions.

Complete step by step solution:
Here we will follow the odd and even functions of the trigonometric functions.
In odd functions, $f( - x) = f(x)$
We have two odd trigonometric functions cosine and sine.
Therefore, referring the trigonometric table for value -
$\cos ( - 45^\circ ) = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$ …. (A)
$\sec ( - 45^\circ ) = \sec 45^\circ = \sqrt 2$ … (B)
Similarly, in even functions, $f( - x) = - f(x)$
We have four even trigonometric functions. Sine, cosec, tangent and cot are even functions.
Therefore, referring the trigonometric table for value -
$\sin ( - 45^\circ ) = - \sin (45^\circ ) = - \dfrac{1}{{\sqrt 2 }}$ …. (c)
$co\sec ( - 45^\circ ) = - co\sec 45^\circ = - \sqrt 2$ … (D)
$tan( - 45^\circ ) = - tan(45^\circ ) = - 1$ …. (E)
$\cot ( - 45^\circ ) = - \cot 45^\circ = - 1$ … (F)
Equations (A), (B), (C), (D), (E) and (F) are the required solutions.

Note : Always remember the trigonometric table for the values of the different angles for all the six trigonometric functions for the efficient and the accurate solution. Also, be careful while simplifying the equations. Also, be wise while opening the brackets and the sign outside the bracket.
Remember the All STC rule, it is also known as ASTC rule in geometry. It states that all the trigonometric ratios in the first quadrant ( $0^\circ \;{\text{to 90}}^\circ$ ) are positive, sine and cosec are positive in the second quadrant ( $90^\circ {\text{ to 180}}^\circ$ ), tan and cot are positive in the third quadrant ( $180^\circ \;{\text{to 270}}^\circ$ ) and sin and cosec are positive in the fourth quadrant ( $270^\circ {\text{ to 360}}^\circ$ ).