Angles Formulas

The angles formulas are used to find the measures of the angles. An angle is formed by two intersecting rays, called the arms of the angle, sharing a common endpoint. The corner point of the angle is known as the vertex of the angle. The angle is defined as the measure of turn between the two lines. Radians or degrees are units of angle. Let us discuss the central angle formulas of the circle, multiple angles formulas, and the double angles formulas in along with a few solved examples.

What Are Angle Formulas?

Here, we have discussed the angle formulas pertaining to the angle at the center of the circle formed between two radii and the arc. Let us also focus on the formulas on multiple angles and double angles expressed in trigonometry.

Multiple Angle Formulas

The multiple angles generally appear in trigonometric functions. The values of multiple angles are not possible to find directly but their values can be calculated by expressing each trigonometric function in its expanded form. These multiple angles of the form sin nx, cos nx, and tan nx are expressed in terms of sin x, and cos x only are derived from the Eulers formula and Binomial Theorem. The following multiple angle formula identities are used in mathematics.

Formula 1: The sin formula for multiple angle is:

Sin nθ = \(\sum_{k=0}^{n}\;cos^{k}\theta \; sin^{n-k}\theta\; Sin\left [\frac{1}{2}\left(n-k\right)\right]\pi\)

where n=1,2,3,……

General formulas are, 

Sin2θ =2 × Cosθ.Sinθ

Sin3θ =3Sinθ - 4Sin³θ

Formula 2: The multiple angle’s Cosine formula is given below:

Cos nθ =\(\sum_{k=0}^{n}cos^{k}\theta \,sin^{n-k}\theta \;cos\left [\frac{1}{2}\left(n-k\right)\pi\right]\)

where n = 1,2,3

The general formula goes as:

Cos2θ = Cos²θ – Sin²θ

Cos3θ = 4Cos³θ – 3Cosθ

Formula 3: Tangent Multiple Angles formula

Tan nθ= Sin nθ/ Cos nθ

Where n = 1,2,3....

Double Angle Formulas

Double angle formulas are used to express the trigonometric ratios of double angles (2θ) in terms of trigonometric ratios of single angle (θ). The double angle formulas are the special cases of (and hence are derived from) the sum formulas of trigonometry and some alternative formulas are derived by using the Pythagorean identities. We derive the double angle formulas of sin, cos, and tan by substituting A = B in each of the above sum formulas. Also, we derive some alternative formulas that are derived using the Pythagorean identities. 

The double angle formulas of sin, cos, and tan are,

• sin 2A = 2 sin A cos A (or) (2 tan A) / (1 + tan²A)
• cos 2A = cos²A - sin²A (or) 2cos²A - 1 (or) 1 - 2sin²A (or) (1 - tan²A) / (1 + tan²A)
• tan 2A = (2 tan A) / (1 - tan²A)

What is Central Angle of Circle Formula?

The central angle of a circle formula calculates the angle between two radii of a circle. A central angle can also be defined as an angle subtended by the arc of a circle and the 2 radii at the center of the circle. The radius vectors form the arms of the central angle. To calculate the central angle, we require the measure of the arc length that subtends the central angle at the center and the radius of the circle. The central angle of a circle formula is given as,

Central angle, θ = (Arc length × 360º)/(2πr) degrees or 

Central angle, θ = Arc length/r radians, where r is the radius of the circle.

Examples Using Angle Formulas

Example 1: Jill has a circular segment with an arc length is 7π and a radius is 9 units. Find the angle of the segment using the angles formulas.

Solution:   

Arc length =  7π (given)                   

Radius = 9 units(given)                  

Using Angles Formulas,                                      

Angle = (Arc Length × 360^o)/2π r

Angle = (7π × 360^o)/2π × 9}                                                                

= 140^o degrees.

Therefore, the angle of the segment is 140^o

Example 2: If tan A = 3 / 4, find the values of sin 2A, cos 2A, and tan 2A.

Solution:

Since the value of tan A is given, we use the double angle formulas for finding each of sin 2A, cos 2A, and tan 2A in terms of tan.

\( \begin{align}\sin 2A &= \dfrac{2 \tan A }{1+\tan^2A}\\[0.2cm] &= \dfrac{2 \left(\dfrac{3}{4} \right)}{1+\left(\dfrac{3}{4} \right)^2}\\[0.2cm]&=\dfrac{24}{25} \end{align}\)

\( \begin{align} \cos 2A &= \dfrac{1-\tan ^{2} A}{1+\tan ^{2} A}\\[0.2cm] &= \dfrac{1-\left(\dfrac{3}{4} \right)^2}{1+\left(\dfrac{3}{4} \right)^2}\\[0.2cm] &=\frac{7}{25} \end{align}\)

\( \begin{align}\tan 2A &= \dfrac{2 \tan A }{1-\tan^2A}\\[0.2cm] &= \dfrac{2 \left(\dfrac{3}{4} \right)}{1-\left(\dfrac{3}{4} \right)^2}\\[0.2cm]&=\dfrac{24}{7} \end{align}\)

Therefore, sin 2A = \(\dfrac{24}{25}\), cos 2A = \(\dfrac{7}{25}\), and tan 2A = \(\dfrac{24}{7}\)

Example 3: Prove that \(\frac{3 \sin\theta -4 sin^{3}\theta}{4 cos^{3}\, \theta – 3 cos\, \theta}=tan\;3\theta\) by using multiple angle formulas.

Solution:

Using the multiple angle formulas,
Sin3θ =3Sinθ - 4Sin³θ, and
Cos3θ =4Cos³θ – 3Cosθ

Putting the values in L.H.S,

\(=\frac{3Sin\theta - 4Sin^3\theta}{4Cos^3\theta – 3Cos\theta}\)
\(=\frac{Sin3\theta}{Cos3\theta}\)
R.H.S = tan3θ

Henced proved \(\frac{3 \sin\theta -4 sin^{3}\theta}{4 cos^{3}\, \theta – 3 cos\, \theta}=tan\;3\theta\).