Angles Formulas

Angle Formulas are the formulas that are used to find the measure of an angle. In geometry, an angle is an essential measurement of a geometric shape. An angle is defined as the degree of rotation about the point of intersection between two lines or planes that are required to bring one into correspondence with the other.

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.

What Are Angle Formulas?

Angle formulas are essential formulas in mathematics, that provide relationships between the angles and sides of triangles. These formulas are crucial for solving various problems in geometry, physics, engineering, and many other fields.

In the later part of the article, we have covered various angle formulas and their respective examples in detail.

Multiple Angle Formulas

Multiple angle formulas used in trigonometry are:

Sin formula for multiple angle is:

sin nθ = ∑ⁿ₀ cos^kθ.sin^(n-k)θ.sin[1/2(n-k)]π

where, n = 1, 2, 3, ...

Cosine formula for multiple angle is:

cos nθ = ∑ⁿ₀ cos^kθ.sin^(n-k)θ.cos[1/2(n-k)]π

where, n = 1, 2, 3, ...

Tangent formula for multiple angle is:

tan nθ = sin nθ/cos nθ

where, n = 1, 2, 3, ...

Double Angle Formulas

Double angle formulas are trigonometric identities that express trigonometric functions of double angles (2θ) in terms of single angles (θ). These formulas are useful for simplifying expressions and solving trigonometric equations.

Double angle formulas of sin, cos, and tan are,

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

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

• tan 2A = (2 tan A) / (1 - tan²A)

Central Angle of Circle Formula

A central angle is an angle whose vertex is the center of the circle and whose sides (or arms) extend to the circumference of the circle, creating two radii. The central angle intercepts an arc on the circle.

Central angle θ (in radians) of a circle can be related to the arc length s and the radius r of the circle using the following formula:

θ = s/r

where:

• θ is the Central Angle in Radians
• s is the Arc Length
• r is the Radius of theCircle

Formula for Finding Angles

There are various types of formulas for finding an angle; some of them are the central angle formula, double-angle formula, half-angle formula, compound angle formula, interior angle formula, etc.

• We use the Central Angle Formula to determine the angle of a segment made in a circle.
• We use the Sum of the Interior Angles formula to determine the missing angle in a polygon.
• We use the Trigonometric Ratios to find the missing angle of a right-angled triangle.
• We use the Law of Sines or the Law of Cosines to find the missing angle of a non-right angle triangle.

Name	Formula	How to Find Unknown Angle?
Central Angle Formula	θ =(s × 360°)/2πr Here, s is the Arc Length r is the Radius of the Circle	Substitute the values of arc length and the radius of the circle to determine the angle of a segment made in a circle.
Sum of Interior angles Formula	180°(n-2) Here, n is the Number of Sides of a Polygon	To determine the unknown interior angle of a polygon, first, calculate the sum of all interior angles using this formula and then subtract the sum of all known angles from the result.
Trigonometric Ratios	sin θ = opposite side/hypotenuse cos θ = adjacent side/hypotenuse tan θ = opposite side/adjacent side	Depending on the available two sides of a right-angled triangle, choose one of these trigonometric ratios to find the unknown angle.
Law of Sines	a/sin A = b/sin B = c/sin C Here, A, B, and C are the Interior Angles of a Triangle a, b, and c are their Respective Opposite Sides	When we know two sides and a non-included angle (or) two angles and a non-included side, then the law of sines can be used to determine the unknown angles of a triangle.
Law of cosines	a2 = b2 + c2 - 2bc cos A b2 = c2 + a2 - 2ca cos B c2 = a2 + b2 - 2ab cos C Here, A, B, and C are the Interior Angles of a Triangle a, b, and c are their Respective Opposite Sides	When we know three sides (or) two sides and an included angle, then the law of cosines can be used to determine the unknown angles of a triangle.

Read More,

• Trigonometric Identities
• Trigonometry Table
• Half Angle Formulae

Examples Using Angle Formulas

Example 1: Find the angle at the vertex B of the given triangle using one of the trigonometric formulae for finding angles.

Solution: 

Given,

• BC = 3 units = Adjacent side of θ
• AC = 4 units = Opposite side of θ

In this case, we know both the opposite and adjacent sides of θ. Hence, we can use the tangent formula to find θ.

⇒ tan θ = opposite side/adjacent side

⇒ tan θ = 4/3

⇒ θ = tan^-1 (4/3) ⇒ θ = 53.1°

Hence, the angle at vertex B is 53.1°.

Example 2: Find the angles at vertices X and Y, if ∠Z = 35° and x = 3 inches, y = 8 inches, and z = 3.5 inches.

Solution:

Given,

∠Z = 35° and x = 6 inches, y = 3 inches, and z = 3.5 inches

Since we know all three sides and an angle, we can use the sine rule formula.

From the sine rule formula, we have

x/sin X = y/ sin Y = z/sin Z

Now,

y/ sin Y = z/sin Z

⇒ 3/sin Y = 3.5/sin 35°

⇒ 3/sin Y = 3.5/0.574    {Since, sin 35° = 0.574}

⇒ sin Y = 3 × (0.574/3.5) = 0.492

⇒ ∠Y = sin⁻¹(0.492) = 29.47°

We know that, sum of three angles in a triangle is 180°.

⇒ ∠X + ∠Y + ∠Z = 180°

⇒ ∠X + 29.47° + 35° = 180°

⇒ ∠X = 180° - 64.47‬° = 115.53°

Hence, ∠X = 115.53‬° and ∠Y = 29.47°

Example 3: Calculate the fifth interior angle of a pentagon if four of its interior angles are 110°, 85°, 136°, and 105°.

Solution:

Number of sides of a pentagon (n) = 5.

Now, the sum of all 5 interior angles of a pentagon = 180 (n -2)° 

= 180 (5 - 2)° = 540°.

The sum of the given 4 interior angles = 110°+ 85°+ 136°+ and 105°= 436‬°.

So, the fifth interior angle = 540° - 436‬° = 104‬°

Thus, the fifth interior angle of a pentagon is 104‬°.

Example 4: Determine the value of y and also the measure of angles in the given figure.

Solution:

From the given figure, we can observe that (4y - 6)‬° and (3y + 5)‬° are complementary angles, i.e., the sum of (4y - 6)‬° and (3y + 5)‬° is 90‬°.

⇒ (4y - 6)‬° + (3y + 5)‬° = 90‬°

⇒ (7y - 1)‬° = 90‬°

⇒ 7y = 90‬° + 1‬° = 91‬°

⇒ y = 91‬°/7 = 13‬°

Now, (4y - 6)° = (4 ×13 - 6)° = (52 - 6)° = 46°

(3y + 5)° = (3 × 13 + 5)° = (39 + 5)° = 44°

Example 5: Find the angle at vertex Q in the given triangle using one of the formulas for finding angles.

Solution:

Given, p = QR = 6 cm,  q = PR = 9 cm, and  r = PQ = 7 cm.

Since we know all three sides and an angle, we can use the cosine rule formula to find the angle vertex Q.

⇒ q² = p² + r² - 2pr cos Q

⇒ 9² = 6² + 7² - 2 (6)(7) cos Q

⇒ 81 = 36 + 49 - 84 cos Q

⇒ 81 = 85 - 84cos Q

⇒84 cos Q = 81 - 85

⇒ 84 cos Q = -4 

⇒ cos Q = -4/84 = -1/21

⇒ ∠Q = cos^-1(-1/21) = 92.72°

Hence, the angle at vertex Q, ∠Q = 92.72°.

Example 6: Calculate the angle of a segment made in a circle if the arc length is 12π and the radius is 9 cm.

Solution: 

Given,

• Arc length= 12π
• Radius (r) = 9 cm

Now, the angle formula is:

⇒ θ = (s×360°)/2πr

⇒ θ = (12π × 360°)/(2π × 5)

⇒ θ =12 ×360°/10

⇒ θ = 240°

Hence, the angle is 240°