The word ‘ Trigonometry ’ is derived from the Greek word and the subject is developed to solve geometric problems involving triangles. It is used to measure the sides of a triangle. An angle is a measure of rotation of a given ray about its initial point and the original ray is called the initial side and the final position of a ray after the rotation of the original ray is called the terminal side. If the rotation of a ray is in an anticlockwise direction, then the angle is a positive angle and if the rotation of a ray in a clockwise direction, then the angle is a negative angle. An angle is said to be positive if the rotation is in anti clockwise direction and if the rotation is in a clockwise direction, then the angle is negative. The two types of conventions used for measuring angles are
 Degree Measure
 Radian Measure.
In degree measurement, consider a unit circle where the angle is said to have a measure of degree and it is denoted by the symbol ‘1^{0 }‘. Each degree(1^{0} ) is divided into 60^{ } minutes (also denoted as 60’ ) and each minute (1’) is subdivided into 60 seconds and it is denoted by 60”. Another unit of measurement of an angle is radian. An angle is said to have a measure of 1 radian in which angle subtended at the centre by an arc length of 1 unit in a unit circle. Generally, trigonometric ratios are represented for the acute angle as the ratio of the sides of a right angle triangle.
Tan 0 Degree Value
In a right angled triangle, the opposite side of the right angle is called the hypotenuse side, the side opposite the angle of interest is called the opposite side and the remaining side is called the adjacent side where it forms a side of both the right angle and the angle of interest.
The tangent function of an angle is equal to the length of the opposite side divided by the length of the adjacent side.
\(\tan \theta =\frac{opposite side}{adjacent side}\)By representing the tangent function in terms of sin and cos function, it is given by
\(\tan \theta =\frac{\sin \theta }{\cos \theta }\)Deriving the Value of Tan Degrees
To find the value of tan 0 degree, use sine function and cosine function. Because tan function is the ratio of sine function and cos function. The tan 0 degree value is given as
\(\tan 0^{\circ}=\frac{\sin 0^{\circ}}{\cos 0^{\circ}}=\frac{0}{1}=0\)Similarly,
tan 30^{0 }= Sin 30^{0} / Cos 30^{0 }= \(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}=\frac{1}{\sqrt{3}}\)
tan 45^{0 }= Sin 45^{0} / Cos 45^{0 }= \(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}=1\)
tan 60^{0 }= Sin 60^{0} / Cos 60^{0 }= \(\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}\)
tan 90^{0 }= Sin 90^{0} / Cos 90^{0 }= 1 / 0 = Not defined
In the same way, we can derive other values of tan degrees like 180^{0}, 270^{0} and 360^{0}. The trigonometry table is given below, which defines all the values of tan along with other trigonometric ratios.
Sample problem:
Question 1 :
Find the value of tan 15^{0}
Solution :
To find : tan 15^{0} degree value
Tan 15^{0} = tan (45^{0} – 30^{0 })
We know that the formula, tan ( AB) = (tan A – tan B) / (1+ tan A tan B)
Now substitute the value of tan 30^{0} and tan 45^{0}
\(\tan 15^{\circ}=\tan (45^{\circ}\tan 30^{\circ})=(\tan 45^{\circ}\tan 30^{\circ})/(1+\tan 45^{\circ}\tan 30^{\circ})\)tan 15^{0} = \((1\frac{1}{\sqrt{3}})/(1+\frac{1}{\sqrt{3}})\)
Therefore, \(\tan 15^{\circ}=(\sqrt{3}1)/(\sqrt{3}+1)\)
Question 2 :
Prove that \(\frac{\sin (x + y)}{\sin (xy)}=\frac{tanx+tany}{tanxtany}\)
Solution:
Given : \(\frac{\sin (x + y)}{\sin (xy)}=\frac{tanx+tany}{tanxtany}\)
L.H.S = \(\frac{\sin (x + y)}{\sin (xy)}=\frac{\sin x\cos y +\cos x\sin y}{\sin x\cos y\cos x\sin y}\)
Now, divide the numerator and denominator by cos x cos y ,we get
= \(\frac{\tan x+\tan y}{\tan x\tan y}\)
= R. H.S
Therefore, L.H.S = R.H.S
\(\frac{\sin (x + y)}{\sin (xy)}=\frac{tanx+tany}{tanxtany}\)Hence proved.
Visit BYJU’S for more information on tangent angle in trigonometry and its related articles, and also watch the interactive videos to clarify the doubts.
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