Question

How do you find the six Trig values of \[ - {450^ \circ }?

Answer 1

Hint: We will put \[ - {450^ \circ }\]directly as the value of theta inside the six trigonometric expressions and we will try to find the value of these values.
The six trigonometric expressions are Sine, Cosine, Tangent, Cotangent, Secant and Cosecant.
And, these are expressed in the following short form:
\[\sin \theta \], \[\cos \theta \], \[\tan \theta \], \[\cot \theta \], \[\sec \theta \], and \[\cos ec\theta \].
Now, the rule of all, sin, cos and tan defines the following points:
\[1.\]The value of all trigs has positive value in the first quadrant.
\[2.\]\[\sin \theta ,\cos ec\theta \] are only positive in the second quadrant.
\[3.\]\[\tan \theta ,\cot \theta \] are only positive in the third quadrant.
\[4.\]\[\cos \theta ,\sec \theta \] are only positive in the fourth quadrant.
Now, we also know that \[\sin ( - \theta ) = - \sin \theta \]and, \[\tan ( - \theta ) = - \tan \theta \].
We also know that:
\[\cos ec\theta = \dfrac{1}{{\sin \theta }}\], \[\cot \theta = \dfrac{1}{{\tan \theta }}\], \[\sec \theta = \dfrac{1}{{\cos \theta }}\] and, \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\].

Complete step by step answer:
First of all we will put \[\theta = {450^ \circ }\] in the above trigonometric values.
So, we can write it as:
\[\sin ( - {450^ \circ }),\cos ( - {450^ \circ }),\tan ( - {450^ \circ }),\cot ( - {450^ \circ }),\sec ( - {450^ \circ }),\cos ec( - {450^ \circ })\].
Now, we will calculate the \[\sin ( - {450^ \circ })\] first.
So, we can write it as following:
\[\sin ( - {450^ \circ }) = - \sin ({450^ \circ })\].
So, if we look at it closely, then we can say that we have taken a full clockwise movement in the quadrant of \[{360^ \circ }\] and then we move another \[{90^ \circ }\]from there.
Which says that we will reach at the end of the first quadrant .
But the first quadrant value of \[\sin ({450^ \circ })\] will be the same as \[\sin ({90^ \circ })\], which is equal to \[1\].
So, \[\sin ( - {450^ \circ }) = - 1\].
Similarly, we will calculate the value of \[\cos ( - {450^ \circ })\].
So, in \[\cos ( - {450^ \circ })\] if we move\[{450^ \circ }\] in the anti-clockwise direction, again we will be left at the end of the first quadrant.
So, the value of \[\cos ( - {450^ \circ })\] is also equal to the value of \[\cos ( - {90^ \circ })\], which is equal to \[0\].
So, we can find the value of \[\tan ( - {450^ \circ })\] using the formula \[\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}\].
So, \[\tan ( - {450^ \circ }) = \dfrac{{\sin ( - {{450}^ \circ })}}{{\cos ( - {{450}^ \circ })}} = \dfrac{{ - 1}}{0} = \infty .\]
Again, we can find the value of \[\cot ( - {450^ \circ })\] using the formula \[\cot \theta = \dfrac{1}{{\tan \theta }}\].
So, \[\cot ( - {450^ \circ }) = \dfrac{1}{{\tan ( - {{450}^ \circ })}} = \dfrac{1}{\infty } = 0.\]
Similarly,
\[\sec ( - {450^ \circ }) = \dfrac{1}{{\cos ( - {{450}^ \circ })}} = \dfrac{1}{0} = \infty .\]
And, \[\cos ec( - {450^ \circ }) = \dfrac{1}{{\sin ( - {{450}^ \circ })}} = \dfrac{1}{{ - 1}} = - 1.\]

Therefore, the six trigonometric (Sin, Cos, Tan, Cot, Cosec and, Sec) of \[ - {450^ \circ }\]are \[ - 1,0,\infty ,0, - 1,\infty .\]

Note: Points to remember: If the given value of theta is equal to the one-fourth of the value of the total measurement of the quadrant, then we can easily use the quadrants to calculate the values of the trigs values.