Angles Formula

An angle is the space formed between two intersecting lines or rays. These two lines are called the arms of the angle, and the point where they meet is known as the vertex. Angles are fundamental elements in geometry and are measured in specific units—most commonly in degrees (°) or radians (rad).

An angle is a shape (as shown in the image added below) formed by two lines or rays that diverge from a common point called a vertex. When two rays are intersected, i.e., when half-lines are projected with a common endpoint, an angle is formed. Now, the common endpoints are called vertices, while the rays are known as the arms.

Geometrically, an angle represents a figure created by two rays starting from a common endpoint. The word angle originates from the Latin word "angulus," which means corner.

In mathematics, there are various formulas used to calculate or transform angles. These include:

Double Angle Formulas

Double-angle formulas are the angle formulas that are derived from the sum formulas of trigonometry and some other formulas by using the Pythagorean identities. The double angle formula is the expression of the trigonometric ratios of double angles (2θ) concerning 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 subtended 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 center and radius of the circle derive the central angle formula. The angle can be measured in degrees or radians.

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

Half-Angle Formula

The Half-Angle Formula allows you to find the trig values of half an angle when the full angle is known.

we explore the half-angle formulas for sine, cosine, and tangent. While the values of trigonometric functions for standard angles like 0°, 30°, 45°, 60°, and 90° are well known from trigonometric tables, determining the exact values of angles such as 22.5°, 15°, and others requires a different approach.

This is where half-angle formulas prove especially useful. They allow us to find precise trigonometric values for non-standard angles and are also invaluable in simplifying expressions and proving trigonometric identities. Derived from the double-angle identities, these formulas are expressed in terms of half-angles such as θ/2, x/2, or A/2​.

Sine Half-Angle Formula

\sin\left(\frac{A}{2}\right) = \pm \sqrt{ \frac{1 - \cos A}{2} }

Cosine Half-Angle Formula

\cos\left(\frac{A}{2}\right) = \pm \sqrt{ \frac{1 + \cos A}{2} }

Tangent Half-Angle Formulas

\tan\left(\frac{A}{2}\right) = \pm \sqrt{ \frac{1 - \cos A}{1 + \cos A} }

Compound Angle Formula

A compound angle is simply the sum or difference of two angles, such as:

• A + B (angle A plus angle B)
• A - B (angle A minus angle B)

Instead of calculating trigonometric values of A + B or A - B using geometry or a calculator, we use compound angle formulas. These formulas allow us to express the sine, cosine, or tangent of a compound angle in terms of the sine, cosine, or tangent of the individual angles A and B.

Sine:

sin ⁡(A ± B) = sin⁡A cos⁡B ± cos⁡A sin⁡B

Cosine:

cos ⁡(A ± B) = cos⁡A cos⁡B ∓ sin⁡A sin⁡B

Tangent:

tan⁡(A ± B) = tanA ± tanB​/1 ∓ tanA tanB

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 Euler's 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

Types of Angles

Various types of angles which are given as:

• Acute Angle: Less than 90°
• Right Angle: Exactly 90°
• Obtuse Angle: Between 90° and 180°
• Straight Angle: Exactly 180°
• Reflex Angle: Between 180° and 360°
• Full Rotation: Exactly 360°

➣ Read in detail - [7 different types of angles]

Solved Question on Angle Formula

Question 1. Find the central angle of the arc of a circle with a radius of 9cm and arc length 4π.

Solution:

Given

The arc length is  4π.
The radius is 9cm

Now,

Central angle(θ)= Arc length×360/2πr
=> 4π×360/2π×9
=>80°

Question 2. Find the central angle of the arc of a circle with a radius of 8cm and arc length 2π.

Solution:

Given

The arc length is  2π.
The radius is 8cm

Now,

Central angle(θ)= Arc length×360/2πr
=> 2π×360/2π×8
=>45°

Question 3. Find the central angle of an arc with a radius of 10cm and an Angle arc length of 5π.

Solution:

Given

The arc length is 5π.
The radius is 10cm.

Now,

Central angle= Arc length/r
=>5π/10
=>π/2

Question 4. Find the central angle of an arc with a radius of 4cm and arc length 8π.

Solution:

Given

The arc length is 8π.
The radius is 4cm.

Now,

Central angle= Arc length/r
=>8π/4
=>2π

Question 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

Question 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

Practice Problems on Angle Formulas

Question 1: In a triangle ABC, angle A is twice angle B, and angle C is 30°. Calculate the measures of angles A and B.

Question 2: Two angles of a triangle are in the ratio 2:3. If the third angle is 70°, find the measures of all angles in the triangle.

Question 3: In a right-angled triangle, one of the acute angles is 38°. What is the measure of the other acute angle?

Question 4: The difference between the complement and supplement of an angle is 50°. Find the angle.

Question 5: Two angles are supplementary. If one angle is 3x and the other is 5x, find the value of x and both angles.

Question 6: In an isosceles triangle, the vertex angle is 40°. Calculate the measures of the base angles.

Question 7: The angles of a quadrilateral are in the ratio 2:3:4:5. Find the measure of each angle.

Question 8: Two angles are complementary. If one angle is 15° more than twice the other, find both angles.

Question 9: In a triangle, one angle is 20° more than the smallest angle, and the largest angle is 40° more than the smallest. Find all angles.

Question 10: The exterior angle of a regular polygon is 24°. How many sides does the polygon have?

Question 11: Two angles form a linear pair. If one angle is 5 times the other, find both angles.

Question 12: In a parallelogram, one angle is 3x and its adjacent angle is (90 - x)°. Find the measures of all angles in the parallelogram.

Conclusion

This article features a brief overview of angles as they are defined in geometry and other branches of mathematics such as trigonometry. It then goes further to define angles, especially as the extent of one line about another especially when the two intersect, and provides various formulas relating to angles. The given text entails the information regarding double angle formulas and also central angle formulas and it also uses multiple angle formulas along with the mathematical expressions of these all.

This knowledge is also illustrated through sample problems, which show how these formulas may be used especially when determining central angles of circles. The article is followed by angle practice problems, and they involve triangles, complementary angles, supplementary angles and polygons. In particular, the content of the section provides an introduction to angle formulas and their use in geometry and trigonometry.