Table of Contents

• Introduction
• Properties of Complementary
• Complementary Angles in Right
• Practical Examples of Complementary
• Complementary vs. Supplementary
• Complementary angles in Quadrilaterals and Polygons
• Complementary Angles in
• Trigonometric Problems with Complementary Angles

Introduction

Complementary angles are two angles whose values add up to 90^o.

Since 90^o measures the right angle, it can be concluded that if two angles are complementary, they form a right angle together. For instance, in the above diagram, the ray has divided a right angle into a pair of complementary angles of 30^o and 60^o.

Complementary angles may or may not share a common ray and vertex.

For instance, in the above diagram, one angle is 25^o and the other is 65^o. When added, the sum of both angles is 90^o,which makes it a right angle.

Properties of Complementary Angles

• The sum of complementary angles in 90^o.
• Since by definition acute angles are less than 90^o, the pair of complementary angles will individually be acute angles.
• Complementary angles can be adjacent or nonadjacent angles. When two angles share a common vertex and side, it is called an adjacent angle. Likewise, when two angles do not share a common vertex and side they are called non-adjacent angles.
• To find the value of a missing angle in complementary angles, the value of the known angle is subtracted from 90^o.

For instance, in the above complementary angle, the value of θ can be calculated as –

Θ = 90^o – 53^o

                = 37^o

• In trigonometry, the sine of one angle is equal to the cosine of its complementary.
• The product of the sine and cosine of complementary angles is always 1.

Complementary Angles in the Right Triangle

Complementary angles are naturally present in a right triangle. In a right-angle triangle, the two acute angles are complementary. As the sum of all interior angles in a triangle is 180^o since the right angle is 90^o, the sum of the remaining two acute angles will be 90^o.

The above figure is a right-angle triangle with ∠ABC = 90^o and ∠ACB = 40^o. Since the sum of all interior angles of a triangle is 180^o,

∠BAC = 180^o – (∠ABC + ∠ACB)

Thus,

∠BAC =180^o – (90^o + 40^o)

           = 180^o – 130^o

           = 50^o

Also, 

∠BAC + ∠ACB = 50^o + 40^o = 90^o

Thus, it can be observed that the sum of the values of the two acute angles in the right-angle triangle is equal to that of the right angle. This proves that the angles ∠BAC and ∠ACB are complementary.

Practical Examples of Complementary Angles

• 

The arms of a clock

• 

Complementary angles on arches.

• 

In pizza slices.

• 

The beams in the bridges are welded to form complementary angles for better balance.

•  

In sports such as basketball and football, there are plays where the angle at which the ball is tackled is integral to the strategy.

Complementary vs. Supplementary Angles

Complementary angles in Quadrilaterals and Polygons

In Euclidean geometry, a quadrilateral is a four-sided 2D figure whose sum of internal angles is 360°.  There are five types of quadrilaterals –

1. Rectangle 
2. Square
3. Parallelogram 
4. Rhombus
5. Trapezium

Complementary angles are observed in quadrilaterals and other polygons. Rectangles and squares have right angles and any line passing diagonally to join two opposite ends will create complementary angles on either side.

In the above diagram of the square, ∠ADBC and ∠CBA are divided by a diagonal ray DB. Since both angles are right angles, the ray DB creates 2 pairs of complementary angles –

• ∠ADB and ∠BDC
• ∠ABD and ∠DBC

Squares and rectangles have right angles, thus, when a diagonal connects the opposite angles, they will have adjacent complementary angles that measure 45^o.

Figures such as the trapezium, parallelogram and rhombus have supplementary angles as they have obtuse angles with at least one angle measuring more than 90^o.

Complementary Angles in Trigonometry

The sine and cosine of angles are ratios of the pairs of sides in right-angled triangles. In a right-angled triangle –

• The sine of an angle is the ratio of the opposite angle to the hypotenuse.
• The cosine of an angle is the ratio of the side adjacent to the angle to the hypotenuse.

Thus, in the ΔABC, ∠B is the right angle. Since the sum of the measures of the three angles must be 180^o,

m∠A + m∠B + m∠C = 180^o

Since, m∠B = 90^o

m∠A + m∠C = 90^o    

Thus, in ΔABC, the trigonometric ratios of complementary angles are –

From the above equations, it can be observed that -

• Sine of an angle = cosine of its complementary angle
• Cosine of an angle = sine of its complementary angle
• Tangent of an angle = cotangent of its complementary angle

In the above triangle –

sin C = cos A = AB/ AC

sin A = cos C = AB/ AC

tan A = cot C = AB/BC

tan C = cot A = BC/AB

sec C = cosec A = AC/BC

sec A = cosec C= AC/AB

Trigonometric Problems with Complementary Angles

Q1. If sec 4θ = cosec (θ – 40^o), where 4θ is an acute angle, find the value of θ.

Solution1:

sec 4θ = cosec (θ – 40^o)

⇒ cosec (90° - 4θ) = cosec (θ - 40°),

[Since sec θ = c0sec (90° - θ)]

⇒ (90° - 4θ) = (θ - 40°)

⇒ -4θ - θ = -40° - 90°

⇒ -5θ = -130°

⇒ θ = 26°

Therefore, θ = 26°

Q2. In any right ΔABC, prove that