In a circle of diameter 40 cm, the length of a chord is 20 cm. Find the length of minor arc of the chord.
Solution and Explanation
Diameter of the circle = 40 cm
∴Radius (r) of the circle = \(\frac{40}{2}\,cm=20\,cm\)
Let AB be a chord (length = 20 cm) of the circle.
In ΔOAB, OA = OB = Radius of circle = 20 cm
Also, AB = 20 cm
Thus, ΔOAB is an equilateral triangle.
\(∴θ = 60° =\frac{\pi}{3}\,radian\)
We know that in a circle of radius r unit, if an arc of length l unit subtends an angle θ radian at the centre, then \(θ=\frac{1}{r}\)
\(=\frac{AB}{20}=\frac{20{\pi}}{3}\,cm\)
Thus, the length of the minor arc of the chord is \(\frac{20{\pi}}{3}\,cm.\)
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Concepts Used:
Trigonometric Functions
The relationship between the sides and angles of a right-angle triangle is described by trigonometry functions, sometimes known as circular functions. These trigonometric functions derive the relationship between the angles and sides of a triangle. In trigonometry, there are three primary functions of sine (sin), cosine (cos), tangent (tan). The other three main functions can be derived from the primary functions as cotangent (cot), secant (sec), and cosecant (cosec).
Six Basic Trigonometric Functions:
- Sine Function: The ratio between the length of the opposite side of the triangle to the length of the hypotenuse of the triangle.
sin x = a/h
- Cosine Function: The ratio between the length of the adjacent side of the triangle to the length of the hypotenuse of the triangle.
cos x = b/h
- Tangent Function: The ratio between the length of the opposite side of the triangle to the adjacent side length.
tan x = a/b
Tan x can also be represented as sin x/cos x
- Secant Function: The reciprocal of the cosine function.
sec x = 1/cosx = h/b
- Cosecant Function: The reciprocal of the sine function.
cosec x = 1/sinx = h/a
- Cotangent Function: The reciprocal of the tangent function.
cot x = 1/tan x = b/a
Formulas of Trigonometric Functions:
