Angle Formula: Definition, Different Types, Solved Example

An angle is the degree of rotation around the point of intersection of two lines or planes that are required to put one in line with the other. In geometrical shapes, it is a critical measurement. Two intersecting rays, known as the angle's arms, share a common terminus to form an angle. The vertex of an angle is the point at which the angle's corner points meet. Angle formulas are used to calculate angle measurements. Angle formulas come in a variety of forms. The double-angle formula, half-angle formula, compound angle formula, internal angle formula, and others are among them. In this article, we will explore the angle formula, its various types, and related sample questions.

Keyterms: Angle, Lines, Planes, Double-angle formula, Half-angle formula, Compound angle formula, Internal angle formula, Vertex, Rays

Read More: Angle Between a Line and a Plane

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What is Angle?

The angle is a shape created by two lines or rays diverging from the vertex, which is the common point. When two rays or half-lines projected with a common termination, cross each other, an angle is generated. The vertex of the angle is the corner points, and the rays are the sides, i.e. the lines are the arms. Angles are measured in radians or degrees.

Angles are defined by three letters with the middle letter indicating where the angle is actually produced. For instance in PQR, Q is the provided angle formed by the lines PQ and QR.

Angles are measured in radians or degrees. A radian is defined as the angle formed by enclosing the radius of a circle around its circumference to form an arc in a circle. Radians and degrees are both used to express the angle between two lines. The whole angle of a circle is 360 degrees, often known as two radians. Angles can be converted from radians to degrees using a formula for conversion. As a result, the degree (°), radians, or gradians are commonly used to describe the angle.

Also Read: Intersecting & Non-Intersecting Lines

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Types of Angle in Geometry

Angles in geometry come in a wide range of shapes and sizes, depending on how they are measured. They are the foundations that lead to the construction of more complicated geometrical figures and shapes.

• Acute Angle

An acute angle is defined as one that is less than 90 degrees and lies between 0 and 90 degrees.

• Obtuse Angle

An obtuse angle is the polar opposite of an acute angle. An obtuse angle is one that is less than 180 degrees but higher than 90 degrees.

• Right Angle

A 90-degree angle is always a right angle. An acute angle is one that is less than 90 degrees, while an obtuse angle is one that is higher than 90 degrees.

• Straight Line

A straight line is the same as a straight angle. The angle formed by a straight line is 180 degrees. When the values of two right angles are added together, the result is 180 degrees.

• Reflex Angle

A reflex angle is any angle that is larger than 180 degrees but less than 360 degrees (which is the same as 0 degrees).

• Full Rotation

Full rotation or full angle refers to an angle of 360 degrees. When one of the arms finishes a complete rotation, a full rotation angle is generated.

There are also some other types of angles found in geometry as well. These are-

• Positive angles are those that are measured from the base in a counterclockwise direction. Positive angles are commonly used to indicate angles in geometry. A positive angle is formed when an angle is drawn in the (+x, +y) plane from the origin.
• Negative angles are those that are measured from the base in a clockwise manner. A negative angle is formed when an angle is drawn from the origin towards the (-x, -y) plane.
• Supplementary angles are formed when the sum of two angles equals 180°, while complementary angles are formed when the sum equals 90°.
• Adjacent angles are formed when any two angles are connected to each other through a common arm, having a common vertex, and the non-common arms exist on either side of the common arms.
• When two lines cross at a single point (called a vertex), the angle formed on either side of the shared vertex is known as a vertical angle or a vertically opposite angle.

Also Read: Linear Pair of Angles

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Angle and Its Formula

Central Angle Formula

The formula for calculating the central angle of a circle calculates the angle between two radii of a circle. A central angle is defined as the angle formed by the arc of a circle and the two radii at the circle's centre. The arms of the central angle are formed by the radius vectors. The measure of the arc length that subtends the central angle at the centre and the radius of the circle are required to compute the central angle. The formula for calculating the central angle of a circle is as follows:

Central angle (θ) = \(\frac{Arc\quad Length\quad x\quad 360}{2\pi r}\)

                             OR

Central angle (θ) = \(\frac{Arc Length}{r}{radians}\)

where r refers to the circle’s radius

Double Angle Formula

The trigonometric ratios of double angles (2) are expressed in terms of trigonometric ratios of single angles using double angle formulas. The Pythagorean identities are used to derive some alternative formulas. The double angle formulas are particular cases of (and so derived from) the sum formulas of trigonometry. By substituting A = B in each of the previous sum formulas, we get the double angle formulas for sin, cos, and tan. In addition, we derive various alternative formulas based on Pythagorean identities.

Double angle formulas for sin, cos, and tan respectively is as follows:

• sin 2A = 2 sin A cos A or,

sin 2A = (2 tan A) / (1 + tan2A)

• cos 2A = cos2A - sin2A or,

cos 2A = 2cos2A - 1 or,

• cos 2A = 1 - 2sin2A or,

cos 2A = (1 - tan2A) / (1 + tan2A)

• tan 2A = (2 tan A)/(1 - tan2A)

Multiple Angle Formula

Multiple angles are most commonly seen in trigonometric functions. Multiple angle values are not immediately determinable, but they can be determined by expressing each trigonometric function in its expanded form. The Eulers formula and the Binomial Theorem are used to calculate numerous angles of the type sin nx, cos nx, and tan nx that are stated in terms of sin x and cos x alone. In mathematics, the following many angle formula identities are employed.

Multiple Angle Formula for Sin

sin nθ = n summation k = 0 coskθ sinn-kθ sin [1/2 ((n-k))] π 

Where n will be from 1, 2, 3…

So, the general formula for sine will be-

 sin 2θ = 2 × cosθ.sinθ

sin 3θ = 3sinθ - 4sin3θ

Multiple Angle Formula for Cosine

cos nθ = n summation k=0 coskθ sinn-kθ cos[1/2 ((n-k))] π

Where n will be from 1, 2, 3…

So, the general formula for cosine will be-

cos2θ = cos2θ – sin2θ

cos3θ = 4 cos3θ – 3cosθ

Multiple Angle Formula for tan

tan nθ= sin nθ/ cos nθ

Where n will be from 1, 2, 3...

Also Read: 

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Things to Remember

• Angles are used in geometry to describe shapes like polygons and polyhedrons, as well as to explain how lines behave.
• A protractor is used to measure the angle in degrees. Angles of 30°, 45°, 60°, 90°, and 180° are shown here. Angle types are determined by angle values expressed in degrees.
• Angles can alternatively be expressed in radians, or in terms of pi (π). So, 180 degrees is equal to 1 radian.
• The measurement of an angle is usually calculated in degrees, radians or gradian.
• The term "zero angle" refers to an angle with a measurement of zero degrees.
• Generally, angle are classified based on rotation and magnitude.