Question

How do you tell whether the value of tan $90$ degrees is positive, negative, zero or undefined?

Answer 1

Hint: In order to solve this question, we refer to the standard angles of trigonometry.
As we know the value of the given term, after using this, we refer to the standard trigonometric table of values to find the value of the respective angles.

Complete step by step solution:
In this question, we are asked to find whether a given trigonometric value is positive, negative, zero or undefined. The given trigonometric value is $\tan {90^ \circ }$
Now as we know that, according to the standard angles of trigonometry, $\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}$
Therefore, $\tan {90^ \circ }$ can also be expressed as below:
$\tan {90^ \circ } = \dfrac{{\sin {{90}^ \circ }}}{{\cos {{90}^ \circ }}}$
Now, let us refer to the standard trigonometric table of values to find the value of the respective angles.

$\theta $	${0^ \circ }$	${30^ \circ }$	${45^ \circ }$	${60^ \circ }$	${90^ \circ }$
$\sin \theta $	$0$	$\dfrac{1}{2}$	$\dfrac{{\sqrt 2 }}{2}$	$\dfrac{{\sqrt 3 }}{2}$	$1$
$\cos \theta $	$1$	$\dfrac{{\sqrt 3 }}{2}$	$\dfrac{{\sqrt 2 }}{2}$	$\dfrac{1}{2}$	$0$
$\tan \theta $	$0$	$\dfrac{{\sqrt 3 }}{3}$	$1$	$\sqrt 3 $	Undefined

Now, according to the table given above:
$\sin {90^ \circ } = 1$
While $\cos {90^ \circ } = 0$
Therefore, $\tan {90^ \circ } = \dfrac{1}{0}$ , since anything divided by zero becomes undefined or infinity.

Therefore the value of tan${90^ \circ }$ is undefined.

Note: Trigonometry is a branch of mathematics which deals with triangles. There are many trigonometric formulas that establish a relation between the lengths and angles of respective triangles. In trigonometry, we use a right-angled triangle to find ratios of its different sides and angles such as sine, cosine, tan, and their respective inverse like cosec, sec, and cot. Some common formulas of trigonometric identities are:
${{sin\theta = }}\dfrac{{{\text{perpendicular}}}}{{{\text{hypotenuse}}}}$ , where perpendicular is the side containing the right angle in a right angled triangle and hypotenuse is the side opposite to the perpendicular.
${{cos\theta = }}\dfrac{{{\text{base}}}}{{{\text{hypotenuse}}}}$ , where base is the side containing the perpendicular and hypotenuse
${{tan\theta = }}\dfrac{{{\text{perpendicular}}}}{{{\text{base}}}}$