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Question: How do you tell whether the value of tan \(90\) degrees is positive, negative, zero or undefined?...

How do you tell whether the value of tan 9090 degrees is positive, negative, zero or undefined?

Explanation

Solution

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∘\tan {90^ \circ }
Now as we know that, according to the standard angles of trigonometry, tan⁡θ=sin⁡θcos⁡θ\tan \theta = \dfrac{{\sin \theta }}{{\cos \theta }}
Therefore, tan⁡90∘\tan {90^ \circ } can also be expressed as below:
tan⁡90∘=sin⁡90∘cos⁡90∘\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.

θ\theta 0∘{0^ \circ }30∘{30^ \circ }45∘{45^ \circ }60∘{60^ \circ }90∘{90^ \circ }
sin⁡θ\sin \theta 0012\dfrac{1}{2}22\dfrac{{\sqrt 2 }}{2}32\dfrac{{\sqrt 3 }}{2}11
cos⁡θ\cos \theta 1132\dfrac{{\sqrt 3 }}{2}22\dfrac{{\sqrt 2 }}{2}12\dfrac{1}{2}00
tan⁡θ\tan \theta 0033\dfrac{{\sqrt 3 }}{3}113\sqrt 3 Undefined

Now, according to the table given above:
sin⁡90∘=1\sin {90^ \circ } = 1
While cos⁡90∘=0\cos {90^ \circ } = 0
Therefore, tan⁡90∘=10\tan {90^ \circ } = \dfrac{1}{0} , since anything divided by zero becomes undefined or infinity.

Therefore the value of tan90∘{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:
sinθ=perpendicularhypotenuse{{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.
cosθ=basehypotenuse{{cos\theta = }}\dfrac{{{\text{base}}}}{{{\text{hypotenuse}}}} , where base is the side containing the perpendicular and hypotenuse
tanθ=perpendicularbase{{tan\theta = }}\dfrac{{{\text{perpendicular}}}}{{{\text{base}}}}