Complementary Angles

Two angles are called complementary ,If the sum of their angles equals 90 degree.

We are aware that an angle is made of of two rays drawn from the same point. If among the two rays one is fixed and the other one is rotated about the fixed point, it forms different angles as follows.

1. Zero Angle: If the two rays coincide initially then the angle formed is 0.
Zero Angle

2. Acute Angle: If the two rays form angle less than 90 then it is said to be acute angle.
Acute Angle

3. Right Angle: If the two rays form 90 then the angles are said to form right angle.
Right Angle


4. Obtuse Angle: If the angle formed by the two rays is above 90° and less than 180° then they are said to form an obtuse angle.
Obtuse Angle


5. Straight Angle: If the two rays are on opposite sides, and form a straight line, then the angle formed is called obtuse angle.
Straight Angle

6. Reflex Angle: If the angle formed by the two rays is above 180 and less than 360 then the angle formed is called Reflex Angle.
Reflex AngleReflex Angle Example


7. Complete Angle: If the ray which is rotating coincide with the fixed ray after completing on rotation, then angle formed is 360°.
Complete Angle

Complementary angles definition: Pair of angles whose sum is 90° is said to be complementary angles.Example: 40° and 50°.
Complementary AnglesComplementary Angles Example


The sum of the angle is 40° + 50° = 90°.

Adjacent complementary angles: Adjacent complementary angles are those which are adjacent and whose sum = 90°.
Adjacent Complementary Angles

Complementary Angles Theorem

If a and b are complement of each other, then sin a = cos b.

This can also be stated as if θ is the given angle, then its complement is ( 90 – θ ) is its complement.
The trigonometric ratio of sin θ is equal to the trigonometric ratio of ( 90 – θ )Let us discuss with the following triangle:
Complementary Angles Theorem


sin θ =$\frac{opposite}{hypotenuse}$=$\frac{BC}{AC}$ -----------------------(1)

cos ( 90 – θ ) = $\frac{adjacent}{hypotenuse}$=$\frac{BC}{AC}$ ------------------------(2)

From the statements (1) and (2) we see that,
sin θ = cos ( 90 – θ ) =
Hence, according to complementary angle theorem

sin ( an angle ) = cos ( its complement )

Complementary angles theorem Examples

Examples: (1) for the two complement angles, 40° and 50° ,
sin 40° = cos ( 90° – 40°) = cos 50°

(2) For two angles 60° and 30° which are complementary angles,
sin 60° = cos 30° = $\frac{\sqrt{3}}{2}$

(3) For the pair of equal angles which are complementary, 45 and 45
sin 45° = cos 45° = $\frac{1}{\sqrt{2}}$

Complementary Angles Examples

Example 1:
What is the complementary angle of 45 degree?

Solution:
Given one angle =45 degree
=>complementary angle =90-45
=>complementary angle =45 degree

Example 2:
The smaller of the two complementary angle is 46 less than the larger angles. Find the measure of the two angles.

Solution:
Let 1 angle = x
and the other = x + 46

Total = x + x + 46 = 90

2x + 46 = 90
2x = 44
x = 22
Hence one angle =22
and another angle = 22 +46=68


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