Value of Tan 15

The value of tan (15°) is 0.26794919243. Tangent (tan) is one of the functions of Trigonometry that deals with the relationships between the angles and sides of a right-angled triangle. So in short, we can say that measuring a triangle (specifically right-angled triangle) is trigonometry. A right-angled triangle is a triangle having one of its interior angles as 90 degrees. Thus, in that case, the three sides of the triangle can be named as:

  • Hypotenuse: The longest side of the triangle which is opposite to 90 degrees.

  • Perpendicular (opposite): It is the side opposite to the unknown angle and perpendicular to the base (that is, the angle between base and perpendicular is 90 degrees).

  • Base (adjacent): It is the side on which triangle rests and it also contains both the angles (90 degrees and unknown angle .

                         

If the unknown angle is 15 degrees then we can find the value of tan 15 degrees.

The most commonly used trigonometric function is sine, cos, and tan. Out of which, only sin and cos are the basic while the rest can be derived from sin and cos.


             \[\sin \left( \theta  \right) = \frac{{opposite}}{{Hypotenuse}}\]                 \[\cos \left( \theta  \right) = \frac{{Adjacent}}{{Hypotenuse}}\]


As mentioned above, tan is also derived from sin and cos. That is, 

 

\[\frac{{\sin \left( \theta  \right)}}{{\cos \left( \theta  \right)}} = \frac{{\frac{{opposite}}{{Hypotenuse}}}}{{\frac{{Adjacent}}{{Hypotenuse}}}} = \frac{{Opposite}}{{Adjacent}} = \tan \left( \theta  \right)\]


\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]


Therefore,


\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]


Now, if the value of is 15 degrees, that means one of the angles of a right-angled triangle is 15 degrees then we can find the value of tan 15 by placing the length of an opposite and adjacent side in the formula. Let us learn how to find the value of the tangent of any angle.

Finding the Tangent

Let us take an example of the right-angled triangle given below. We can see two unknown angles of less than 90 degrees (A and B). If we consider angle A as the unknown angle then 

side measuring 5 will be hypotenuse, side measuring 3 will be base, side measuring 4 will be perpendicular. Base and Perpendicular will reverse if we take angle B as .


So according to the formula of tan that is,


\[\tan \left( \theta  \right) = \frac{{Opposite}}{{Adjacent}}\]

We can say that the tangent of angle B is the ratio of side measuring 3 over the side measuring 4 or (3/4=0.75) and the tangent of angle A, is the ratio of side measuring 4 over the side measuring 3 or (4/3=1.33).


Similarly, we can find the value of tan 15, if one of the unknown angles of a right-angled triangle is 90 degrees. The approximate value of tan 15° would be 0.269.


Finding tan 15 value without using sides:


Let us take a right angle triangle having 15 degrees as one of its angles. So, if in a right-angled triangle two angles are 90 and 15 degrees respectively, then the third angle should be:

180 - (90 + 15) = 75 degrees. The length of the base is x, perpendicular is y and hypotenuse is z.

                         

To find tan 15 degree value we can represent 15 as 45 - 30.

tan(15°) = tan(45°-30°)


According to the formula of tan(A - B):

tan(A - B) = (tanA - tanB) /(1 + tan A tan B)

 

⇒tan(45°-30°) = (tan45°- tan30°)/(1+tan45°tan30°)

                        = {1- (1/√3)} / {1+(1*1/√3)}

∴ tan (15°) = (√3 - 1) / (√3 + 1)

   tan (15°) = 2 - √3

On simplification, we get 0.26794919243.

Therefore, 

\[{\mathbf{The}}{\text{ }}{\mathbf{value}}{\text{ }}{\mathbf{of}}{\text{ }}{\mathbf{tan}}{\text{ }}\left( {{\mathbf{15}}^\circ } \right){\text{ }}{\mathbf{is}}{\text{ }}{\mathbf{0}}.{\mathbf{26794919243}}\]



Finding the value of Tan 15 degrees using Sin and Cos

Let us take a right angle triangle having 15 degrees as one of its angles. If the value of sin 15 and cos 15 is given then we can find the value of tan 15.

According to the Formula, 

\[\tan \theta  = \frac{{\sin \left( \theta  \right)}}{{\cos \left( \theta  \right)}}\]

\[Tan{\text{ }}\left( {15^\circ } \right){\text{ }} = \;\frac{{Sin{\text{ }}15}}{{Cos{\text{ }}15}}\].

\[Sin{\text{ }}15^\circ {\text{ }} = {\text{ }}sin{\text{ }}\left( {45{\text{ }}-{\text{ }}30} \right)^\circ {\text{ }}and{\text{ }}cos{\text{ }}15{\text{ }} = {\text{ }}cos{\text{ }}\left( {45{\text{ }}-{\text{ }}30} \right)^\circ \]

\[\therefore \tan \left( {{{15}^o}} \right) = \frac{{\sin {{\left( {45 - 30} \right)}^o}}}{{\cos {{\left( {45 - 30} \right)}^o}}}\]

Trigonometry formulas states that,

sin(A – B) = sin A cos B – cos A sin B         and

cos (A – B) = cos A cos B + sin A sin B

Therefore,

\[\tan \left( {{{15}^o}} \right) = \frac{{{{\left( {\sin {{45}^o}\cos {{30}^o} - \cos {{45}^o}\sin {{30}^o}} \right)}^o}}}{{\left( {\cos {{45}^o}\cos {{30}^o} + \sin {{45}^o}\sin {{30}^o}} \right)}}\]


On putting the values of sin 30°, sin 45°, cos 30° and cos 45°, we get,

tan (15°)=  (1/√2.√3/2 – 1/√2.½) / (1/√2.√3/2 + 1/√2.½)

tan 15° = √3 – 1/ √3 + 1

∴ tan (15°) = (√3 - 1) / (√3 + 1)

   tan (15°) = 2 - √3

On simplification, we get 0.26794919243.

Therefore, 

The value of tan (15°) is 0.26794919243

 

Using the Tangent to Find a Missing Side

The side of the right-angled triangle using tan can be found out by placing the values in the formula. For example: A right-angled triangle with one of the angles as 66 degrees. If the length of the perpendicular is given then the length of the base can be found out. Let us take base as x.  

We know, 

                  Tan= Opposite/Adjacent

           Tan 66 = 5/x

On solving the equation,

              tangent

Therefore, by using the value of tan 66 we can find the length of the base. Similarly, we can find the length of the perpendicular if the length of the base is given. Tan has its application in the real world - it is used in construction companies to construct buildings on hills, etc.


FAQ (Frequently Asked Questions)

Q1: What are the Practical Applications of Tan?

Ans: Tan is used in the concept of height and distance which finds its applications in various fields like:

  1. To Find the Slope and Height During Construction: We can find the height of a building or monument if the angle of elevation and the distance of the building is given.

  2. The Use of Trigonometry to Measure the Height of a Mountain or a Building: Basically, the height of the mountain or building can be easily measured using the trigonometric ratios. 

  3. To Calculate the Angle of Elevation or Depression Required to Reach the Target: Incase of aviation, wind plays a most important role. In aviation, the angle of depression and the angle of elevation is used depending on the case.

Q2: What is Pythagoras Theorem?

Ans: Pythagoras theorem is applicable only on a right-angled triangle. It explains that in a right-angle triangle, the square of the hypotenuse will always be equal to the sum of the squares of the opposite side and the square of the adjacent side. 

Pythagoras Theorem formula;

(Hypotenuse)2=(Base)2+ (Perpendicular)2