About Value of Tan 15
This article will explain how to calculate 15 degrees of tan using trigonometry formulas. However, let us first get acquainted with the concepts of trigonometry. An angle or length in a right triangle is trigonometry. In trigonometry, lengths and angles are related to each other. The study of trigonometry involves studying the right triangle and is derived from three Greek letters, “trigono” means three and “gon” means sides, and meter means to measure.
A trigonometric ratio consists of Sine, Cosine, and Tangent. This trigonometric ratio is taken into account when designing trigonometric functions, identities, and formulas. We will use one of these trigonometric formulas to find the tangent value of 15 degrees. In addition to finding tan 15, we will also use the sine and cosine ratios as well as a trigonometry table to find the value of tan 15.
Knowing sin (15°) and cos (15°) will tell us how to find Tan (15°). When a right angle triangle is constructed, the tangent equals the ratio between the sine and cosine functions of the same angle. Thus, once we find the values for sin 15° and cos 15°, we can easily determine the tangent value. The next step in this article is to find tan 15's value.
The value of tan (15°) is 0.26794919243. A right-angled triangle has angles that are determined by angles of a right-angled triangle, and sides that are determined by sides of the right-angled triangle. Thus, trigonometry can be defined as the measurement of triangles (specifically right-angled triangles). If one of the interior angles of a triangle is 90 degrees, the triangle is right-angled. Therefore, the three sides of the triangle can be written as follows:
Hypotenuse: The side opposite to the 90-degree mark on a triangle.
Perpendicular (opposite): It is perpendicular to the base and opposite to the unknown angle (that is, the angle between the perpendicular and the base is 90 degrees).
Based (adjacent): It contains both the 90 degree and unknown angle angles as well as the triangle.
Finding the Tangent
We will use the right-angled triangle shown below as an example. There are two unknown angles of less than 90 degrees (A and B). In the example above, angle A is the unknown angle, so the side measuring 5 is the hypotenuse, the side measuring 3 is the base, and the side measuring 4 is the perpendicular. If we consider angle B as , then Base and Perpendicular will be reversed.
So according to the formula of tan that is,
\[tan\Theta = \frac{Opposite}{Adjacent}\]
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).
A right-angled triangle has an unknown angle of 90 degrees, so we can find the value of tan 15 if that angle is also unknown. Tan 15° is approximately 0.269 degrees.
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°) = (tan 45°- tan30°)/(1+tan 45°tan30°right-angle)
= {1- (1/√3)} / {1+(1*1/√3)}
∴ 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
FAQs on Value of Tan 15
1. What are the Practical Applications of Tan?
Tan is used in the concept of height and distance which finds its applications in various fields like:
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.
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.
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.
2. What is Pythagoras Theorem?
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 |
3. How do you define a Tangent Function?
The tangent function is one of the six fundamental trigonometric functions and is commonly represented by the symbol tan x. This is the ratio between the opposite and adjacent sides of an angle in a right-angled triangle. The tangent function is a trigonometric identity that can be derived from various formulas using different trigonometric identities. The formula for the period of the tangent function f(x) = a tan (bx), is given by, Period = π/|b|. Tangent function tan x is a periodic function and has a period of π/1 = π (Because b =1 in tan x).
4. How many values can tan take?
It is important to memorize a few sine and cosine values, based on triangles of dimensions 30°-60°-90° and 45°-45°-90°. These can be used to calculate the tangent values. tan(90°)=10=undef.