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Angles Formula

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Angle is a space acquired between two intersecting lines. The angles are formed between the two lines known as arms and the point where an angle is formed is known as the vertex. The angle has its own unit of measurement, an angle can be measured in degree or radian. An angle is purely a geometrical structure formed by the intersecting rays and the term itself comes from the Latin word ‘angulus’ which literally means corner. 

What is an Angle formula?

.As studied above we know that the angles are the geometrical structures formed by two intersecting rays and consist of arms and a vertex. Hence, to determine the measure of these angles, the angles formula is used.

In mathematics, there are different formulas involved for the measure of angles and some of them are double angle formula, half-angle formula, compound angle formula, multiple angle formula, and so on.

Double Angle Formulas

Double angle formulas are the angle formula that is derived from the sum formulas of trigonometry and some other formulas by using the Pythagorean identities. The double angle formula is basically the expression of the trigonometric ratios of double angles (2θ) with respect to the trigonometric ratios of single angles (θ).

Mathematical Double angle formulas for sine, cosine, and tangent are given as

=>sin2A=2.sinA.cosA

             (Or)

sin2A=(2 tanA)/(1+tan2A)

=>cos2A=cos2A-sin2A

(Or)

cos2A=2cos2A-1

(Or)

cos2A=(1-tan2A)/(1+tan2A)

=>tan2A=(2tanA)/(1-tan2A)

Central Angle formula

.A central angle is the angle suspended by the arc and the two radii of the circle at the center. The central angle formula is used to determine the angle between the two radii of the given circle. The central angle formula is derived by the center and radius of the circle. The angle can be measured in degree or radian.

Mathematically the central angle formula is given by

In degree

Central angle(θ)= Arc length×360/2πr

where,

r is the radius of the circle

In radian

Central angle(θ)=Arc length/r

where,

r is the radius of the circle

Multiple Angles Formula 

The multiple angle formula is the angle formula that is generally applied for trigonometric functions. The multiple angles formula helps to find the value of multiple angles by expressing the trigonometric functions in expanded forms.

The multiple angles and their trigonometric functions are derived from the Eulers formula and are expressed in the forms of sinx and cosx. There are sine formula, cosine formula, and the tangent formula in the multiple angles formula whose mathematical expressions are given below:

Sine Formula

sin\ n\theta=\sum^n_{k=0}\theta sin^{n-k}\theta sin[1/2(n-k)]\pi

The general sine formulas for multiple angles are:

  • sin2θ = 2.cosθ.sinθ
  • sin3θ = 3.sinθ−4sin3θ

where,

n is the integers

Cosine Formula

cos\ n\theta=\sum^n_{k=0}\theta sin^{n-k}\theta cos[1/2(n-k)]\pi

The general cosine formula for multiple angles are:

  • cos2θ = cos2θ−sin2θ
  • cos3θ = 4cos3θ−3cosθ

where,

n is the integers

Tangent Formula

tan\ n\theta=\frac{sinn\theta}{cosn\theta}

where,

n is the integers

Sample Problems

Problem 1. Find the central angle of the arc of a circle with a radius of 9cm and arc length 4Ï€.

Solution:

Given

The arc length is  4Ï€.

The radius is 9cm

Now,

Central angle(θ)= Arc length×360/2πr

=> 4π×360/2π×9

=>80°

Problem 2. Find the central angle of the arc of a circle with a radius of 8cm and arc length 2Ï€.

Solution:

Given

The arc length is  2Ï€.

The radius is 8cm

Now,

Central angle(θ)= Arc length×360/2πr

=> 2π×360/2π×8

=>45°

Problem 3. Find the central angle of an arc with a radius of 10cm and arc length 5Ï€.

Solution:

Given

The arc length is 5Ï€.

The radius is 10cm.

Now,

Central angle= Arc length/r

=>5Ï€/10

=>Ï€/2

Problem 4. Find the central angle of an arc with a radius of 4cm and arc length 8Ï€.

Solution:

Given

The arc length is 8Ï€.

The radius is 4cm.

Now,

Central angle= Arc length/r

=>8Ï€/4

=>2Ï€

Problem 5. Find the value of sin2A if tanA=1/2.

Solution:

Given 

tanA=1/2

Now

sin2A=2tanA/1+tan2A

=>2(1/2)/1+(1/2)2

=>4/5

Problem 6. Find the value of tan2A if tan=3/5

Solution:

Given

tanA=3/5

Now

tan2A=2tanA/1-tan2A

=>2(1/4)/1-(1/4)2

=>8/15



Last Updated : 11 Jan, 2024
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