Trigonometric Ratios of Complementary Angles

Trigonometry Complementary Angles:

A right triangle is a triangle in which one of the angles is equal to a right angle i.e. 900. In a right triangle, the side opposite to the right angle is the longest side and is called the hypotenuse of the triangle. The side adjacent to the right angle which is horizontal to the plane is called the base. The side which is at 900 to the base is called the perpendicular. In any right triangle, the sum of the other two angles except the right angle is equal to 900. All right triangles obey the Pythagorean theorem which states that “the square on the longest side is equal to the sum of the squares on the other two sides”.

Trigonometric Complementary Ratios:

\[Sin\left( {{{90}^0} - A} \right) = Cos{\text{ }}A\]

\[Cos\left( {{{90}^0} - A} \right) = SinA\]

\[Tan\left( {{{90}^0} - A} \right) = CotA\]

\[Cot\left( {{{90}^0} - A} \right) = TanA\]

\[Sec\left( {{{90}^0} - A} \right) = CosecA\]

\[Cosec\left( {{{90}^0} - A} \right) = SecA\]


Trigonometric Ratios:

Trigonometry is a branch of Mathematics which deals with the measurements of sides and angles of a triangle, in particular a right triangle. There are six trigonometric ratios which represent the angles of a right triangle in terms of their sides. In every right triangle, there are two angles which are not 900. If in a right triangle, any angle that is not a right angle is considered as angle ‘A’, the side adjacent to angle ‘A’ which is not a hypotenuse is called the adjacent side or base and the side opposite to angle ‘A’ is called the opposite side or perpendicular. 


The six trigonometric ratios of angle A is given as:


  1. Sine of an angle ‘A’ is given as 

\[\sin A = \frac{{opposite}}{{Hypotenuse}} = \frac{{BC}}{{AC}}\]

  1. Cosine of an angle ‘A’ is given as

\[\cos A = \frac{{Adjacent}}{{Hypotenuse}} = \frac{{AB}}{{AC}}\]

  1. Tangent of an angle ‘A’ is given as

\[\tan A = \frac{{Opposite}}{{Adjacent}} = \frac{{BC}}{{AB}} = \frac{{\sin A}}{{\cos A}}\]

  1. Cosecant of an angle ‘A’ is given as

\[\cos ecA = \frac{{Hypotenuse}}{{Opposite}} = \frac{{AC}}{{BC}} = \frac{1}{{\sin A}}\]

  1. Secant of an angle ‘A’ is written as

\[\sec A = \frac{{Hypotenuse}}{{Adjacent}} = \frac{{AC}}{{AB}} = \frac{1}{{\cos A}}\]

  1. Cotangent of an angle ‘A’ is written as

\[\cot A = \frac{{Adjacent}}{{Oposite}} = \frac{{AB}}{{BC}} = \frac{{\cos A}}{{\sin A}} = \frac{1}{{\tan A}}\]


From the above definitions, it can be inferred that sine and cosine of any angle are the two fundamental trigonometric ratios from which all the other trigonometric ratios can be defined. 

  • Tangent of an angle is the ratio of its sine to its cosine.

  • Cosecant of an angle is the multiplicative inverse of its sine.

  • Secant of an angle is the multiplicative inverse of its cosine.

  • Cotangent of an angle is the multiplicative inverse of its tangent. Cotangent of an angle can also be defined as the ratio of its cosine to its sine. 


The variation of adjacent and opposite sides based on the reference angle considered is shown in the figure below:


Derivation of Trigonometric Ratios of Complementary Angles:

Any two angles are said to be complementary if their sum is equal to 900. So, complement of any angle is the value obtained by subtracting it from 900. In a right triangle, the sum of the other two angles except the right angle is equal to 900. Therefore, these two angles are considered to be the complementary angles. 


To derive the trigonometric ratios of complementary angles formula, let us consider a right triangle ABC right angled at B. If angle at “C” is taken as the reference angle ‘θ’, then the other reference angle at  ‘A’ is the complement of angle at C. i.e. angle at ‘A’ = 900 - θ.

When ‘θ’ is taken as the reference angle, the opposite side is ‘AB’ and the adjacent side is ‘BC’. AC is opposite to the right angle of the right triangle and hence it is the hypotenuse. The trigonometric ratios of reference angle ‘θ’ is given as


Sine of an angle ‘θ’ is given as 

\[\sin \theta  = \frac{{opposite}}{{Hypotenuse}} = \frac{{AB}}{{AC}} \to \left( 1 \right)\]


Cosine of an angle ‘θ’ is given as

\[\cos \theta  = \frac{{Adjacent}}{{Hypotenuse}} = \frac{{BC}}{{AC}} \to \left( 2 \right)\]


Tangent of an angle ‘θ’ is given as

\[\tan \theta  = \frac{{Opposite}}{{Adjacent}} = \frac{{AB}}{{BC}} \to \left( 3 \right)\]


Cosecant of an angle ‘θ’ is given as

\[\cos ec\theta  = \frac{{Hypotenuse}}{{Opposite}} = \frac{{AC}}{{AB}} \to \left( 4 \right)\]


Secant of an angle ‘θ’ is written as

\[\sec \theta  = \frac{{Hypotenuse}}{{Adjacent}} = \frac{{AC}}{{BC}} \to \left( 5 \right)\]


Cotangent of an angle ‘θ’ is written as

\[\cot \theta \frac{{Adjacent}}{{Oposite}} = \frac{{BC}}{{AB}} \to \left( 6 \right)\]


Now, let us consider the reference angle as angle at ‘A’ which is the complement of angle θ. In this case, the adjacent side is AB and the opposite side is BC. AC remains the hypotenuse. So, the trigonometric ratios of reference angle 900 - θ (i.e. Complementary Ratios) is given as:


Sine of an angle ‘\[9{0^O} - \theta \]’ is given as 

\[\sin \left( {9{0^O} - \theta } \right) = \frac{{opposite}}{{Hypotenuse}} = \frac{{BC}}{{AC}} \to \left( 7 \right)\]


Cosine of an angle ‘\[9{0^O} - \theta \]’ is given as

\[\cos \left( {9{0^O} - \theta } \right) = \frac{{Adjacent}}{{Hypotenuse}} = \frac{{AB}}{{AC}} \to \left( 8 \right)\]


Tangent of an angle ‘’ \[9{0^O} - \theta \] is given as

\[\tan \left( {9{0^O} - \theta } \right) = \frac{{Opposite}}{{Adjacent}} = \frac{{BC}}{{AB}} \to \left( 9 \right)\]


Cosecant of an angle ‘\[9{0^O} - \theta \]’ is given as

\[\cos ec\left( {9{0^O} - \theta } \right) = \frac{{Hypotenuse}}{{Opposite}} = \frac{{AC}}{{BC}} \to \left( {10} \right)\]


Secant of an angle ‘\[9{0^O} - \theta \]’ is written as

\[\sec \left( {9{0^O} - \theta } \right) = \frac{{Hypotenuse}}{{Adjacent}} = \frac{{AC}}{{AB}} \to \left( {11} \right)\]


Cotangent of an angle ‘\[9{0^O} - \theta \]’ is written as

\[\cot \left( {9{0^O} - \theta } \right)\frac{{Adjacent}}{{Oposite}} = \frac{{AB}}{{BC}} \to \left( {12} \right)\]


The inference made by comparing the trigonometric ratios of reference angle ‘θ’ and its complement ‘90o θ’ can be summarized as follows: 


Inference

Reason

\[Sin\left( {{{90}^0} - A} \right) = Cos{\text{ }}A\]

Comparing equations (1) and (7)

\[Cos\left( {{{90}^0} - A} \right) = SinA\]

Comparing equations (2) and (8)

\[Tan\left( {{{90}^0} - A} \right) = CotA\]

Comparing equations (3) and (9)

\[Cot\left( {{{90}^0} - A} \right) = TanA\]

Comparing equations (4) and (10)

\[Sec\left( {{{90}^0} - A} \right) = CosecA\]

Comparing equations (5) and (11)

\[Cosec\left( {{{90}^0} - A} \right) = SecA\]

Comparing equations (6) and (12)


The above table gives the trigonometric ratios of complementary angles formula.

Example Problems:

  1. Without trigonometric tables evaluate \[Sin{35^0}.{\text{ }}Sec{55^0} - \;tan{25^0}.{\text{ }}tan{65^0}.\]


Solution: Using the concept of trigonometry complementary angles, 

\[Sin{35^0}.{\text{ }}Sec{55^0} - \;tan{25^0}.{\text{ }}tan{65^0}.\]

\[ = Sin{35^0}.Cosec\left( {{{90}^0} - {{35}^0}} \right) - \;tan{25^0}.Cot\left( {{{90}^0} - {{65}^0}} \right)\]

\[ = Sin{35^0}.Cosec{35^0} - \;tan{25^0}.Cot{25^0}\]

\[ = 1 - 1 = 0\]

  1. Prove that \[2\left( {Cosec{{60}^0}.Cos{{30}^0}} \right) + \left( {tan{{45}^0}.Cot{{45}^0}} \right) = 3\]without using the standard trigonometric table.


Solution:

Using trigonometry complementary angles formulae,

LHS = 2 (Cosec 600 . Cos 300) + (tan 450 . Cot 450)

= 2 (Cosec 600 . Sin (900 -300) + 1

= 2 (1) + 1

= 2 + 1 = 3 = RHS

Fun Facts:

  • Trigonometric ratios of 450 and its complement are always the same because complement of 450 is also equal to 450.

  • The fact that the relation between 300 and 600 is that they are complementary angles with respect to each other is used to determine the values of trigonometric complementary ratios of standard angles 300 and 600.

FAQ (Frequently Asked Questions)

1. Where are trigonometric ratios of complementary angles worksheets used in real life?


  • Trigonometric ratios are used in measurements of heights and distances in general. 

  • In astronomy, trigonometric ratios are most commonly used to determine the positions of sun, moon, planets and other celestial objects.

  • Trigonometric ratios are also used in locating the latitudes and longitudes during sea voyages and air travel.

  • Trigonometric ratios of complementary angles worksheets are also used to determine the lengths and areas of terrestrial regions. 

  • Complementary ratios are also used in optics and acoustic calculations. 

2. Distinguish the relation between trigonometric ratios of complementary and supplementary angles.

Complementary angles are the angles whose sum is equal to a right angle. (i.e. 900).

Supplementary angles are the angles which add up to give the sum of a straight angle i.e. 1800.


Relation between trigonometric ratios of complementary and supplementary angles are given as follows.