Question

How do you find the six trigonometric functions of 315 degrees?

Answer 1

Find the value of $\sin \left( {{{315}^ \circ }} \right)$ by using $\sin \left( {{{360}^ \circ } - {{45}^ \circ }} \right)$ and $\cos \left( {{{315}^ \circ }} \right)$ by using $\cos \left( {{{360}^ \circ } - {{45}^ \circ }} \right)$. Then, use the quadrant rules for finding which function lies in which quadrant so that we can put positive and negative signs correctly for every function. Other four trigonometric functions can be found by using sine and cosine functions.

Complete Step by Step Solution:
First of all, we will study the quadrant rules of the trigonometric functions which tells us that which function lies in which quadrant so that we put negative and positive signs properly before that function.
Quadrant 1 – In this quadrant all trigonometric functions are positive. This quadrant lies in the angle less than ${90^ \circ }$ and more than 0
Quadrant 2 – In this quadrant, the trigonometric functions sin and cosec are positive while cos, sec, tan and cot functions are negative. This quadrant lies in the angle less than ${180^ \circ }$ and more than ${90^ \circ }$
Quadrant 3 – This quadrant has tan and cot functions as positive and cos, sin, sec and cosec functions negative. This quadrant lies in the angle less than ${270^ \circ }$ and more than ${180^ \circ }$.
Quadrant 4 – cos and sec functions are positive in this quadrant and sin, cosec, tan and cot functions are negative in this quadrant. This quadrant lies in the angle less than ${360^ \circ }$ and more than ${180^ \circ }$.
Now, we have to find the values of all trigonometric functions of 315 degrees, therefore, sine function can be written as –
$\Rightarrow \sin \left( {{{315}^ \circ }} \right) = \sin \left( {{{360}^ \circ } - {{45}^ \circ }} \right)$ , this function does not lie in the fourth quadrant therefore, it can be written as –
$\Rightarrow - \sin \left( {{{45}^ \circ }} \right) = - \dfrac{1}{{\sqrt 2 }}$
Similarly, for cosine function, it can be written as –
$\Rightarrow \cos \left( {{{315}^ \circ }} \right) = \cos \left( {{{360}^ \circ } - {{45}^ \circ }} \right)$ , this function lies in the fourth quadrant therefore, it can be written as –
$\Rightarrow \cos \left( {{{45}^ \circ }} \right) = \dfrac{1}{{\sqrt 2 }}$
Now, we know that,
$\Rightarrow \tan \left( {{{315}^ \circ }} \right) = \dfrac{{\sin \left( {{{315}^ \circ }} \right)}}{{\cos \left( {{{315}^ \circ }} \right)}} = \dfrac{{ - \dfrac{1}{{\sqrt 2 }}}}{{\dfrac{1}{{\sqrt 2 }}}} = - 1$
$\Rightarrow \cos ec\left( {{{315}^ \circ }} \right) = \dfrac{1}{{\sin \left( {{{315}^ \circ }} \right)}} = \dfrac{1}{{ - \dfrac{1}{{\sqrt 2 }}}} = - \sqrt 2$
$\Rightarrow sec\left( {{{315}^ \circ }} \right) = \dfrac{1}{{\cos \left( {{{315}^ \circ }} \right)}} = \dfrac{1}{{\dfrac{1}{{\sqrt 2 }}}} = \sqrt 2$
$\Rightarrow \cot \left( {{{315}^ \circ }} \right) = \dfrac{1}{{\tan \left( {{{315}^ \circ }} \right)}} = \dfrac{1}{{ - 1}} = - 1$

Note:
Don’t go for calculating the exact value by using any other identity as this method gives the most efficient value. Use the quadrant rule to find the value of any trigonometric function correctly as it tells us whether the value of that trigonometric function will be positive or negative.