How to Find Trigonometric Ratios for Complementary Angles
A triangle can be defined as a geometric figure which consists of three sides. There are different types of triangles. In a right angle triangle, one of the three sides is a right angle i.e. 90 degrees. While the side adjacent to the right angle that is horizontal to the plane is called the base.
The side which is at 900 to the base of the triangle is known as the perpendicular. In any right-angle triangle, the sum of the other two angles other than the right angle is equal to 900. All right angle triangles obey the Pythagorean theorem that states that “the square on the hypotenuse is equal to the sum of the squares on the other two sides”.
Trigonometric Ratios
Trigonometry is a branch of mathematics that deals with the measurements of sides and angles of a triangle, in particular a right triangle. There are six trigonometric ratios that represent the angles of a right triangle in terms of its sides. In every right triangle, there are two angles that 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:
Sine of an angle ‘A’ is given as
\[\sin A = \frac{{opposite}}{{Hypotenuse}} = \frac{{BC}}{{AC}}\]
Cosine of an angle ‘A’ is given as
\[\cos A = \frac{{Adjacent}}{{Hypotenuse}} = \frac{{AB}}{{AC}}\]
Tangent of an angle ‘A’ is given as
\[\tan A = \frac{{Opposite}}{{Adjacent}} = \frac{{BC}}{{AB}} = \frac{{\sin A}}{{\cos A}}\]
Cosecant of an angle ‘A’ is given as
\[\cos ecA = \frac{{Hypotenuse}}{{Opposite}} = \frac{{AC}}{{BC}} = \frac{1}{{\sin A}}\]
Secant of an angle ‘A’ is written as
\[\sec A = \frac{{Hypotenuse}}{{Adjacent}} = \frac{{AC}}{{AB}} = \frac{1}{{\cos A}}\]
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.
The tangent of an angle is the ratio of its sine to its cosine.
The cosecant of an angle is the multiplicative inverse of its sine.
The secant of an angle is the multiplicative inverse of its cosine.
The cotangent of an angle is the multiplicative inverse of its tangent. The cotangent of an angle can also be defined as the ratio of its cosine to its sine.
Derivation of Trigonometric Ratios of Complementary Angles
Any two angles are said to be complementary if their sum is equal to 900. So, the 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 complementary angles.
To derive the trigonometric ratios of the complementary angles formula, let us consider a right triangle ABC right angled at B. If the angle at “C” is taken as the reference angle ‘θ’, then the other reference angle at ‘A’ is the complement of the 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’ are 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 the 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:
The above table gives the trigonometric ratios of the complementary angles formula.
Trigonometric Complementary Ratios
According to the trigonometric complementary ratio theorem, the trigonometric function of a complementary angle is defined as another trigonometric function of the original angle. So,
Sin (900 - A) = Cos A
Cos (900 – A) = Sin A
Tan (900 - A) = Cot A
Cot (900 - A) = Tan A
Sec (900 - A) = Cosec A
Cosec (900 - A) = Sec A
Fun Facts
Trigonometric ratios of 450 and its complement are always the same because the 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.
Conclusion
This is all about the trigonometric ratios of the complementary angles. Learn how these ratios are defined and determined in order to utilize the formulas to solve problems. This topic will help you develop your conceptual foundation of trigonometry in a better way.
FAQs on Trigonometric Ratios of Complementary Angles Made Simple
1. What are trigonometric ratios of complementary angles?
Trigonometric ratios of complementary angles describe the relationship between the trigonometric functions of two angles that add up to 90 degrees. In a right-angled triangle, the two acute angles are complementary. The rule states that the trigonometric ratio of an angle is equal to the co-ratio of its complementary angle. For instance, the sine of an angle equals the cosine of its complement.
2. What is the main rule or formula set for complementary angles in trigonometry?
The core rule for complementary angles connects trigonometric functions with their 'co-functions'. For any acute angle A, the fundamental identities are:
- sin (90° – A) = cos A
- cos (90° – A) = sin A
- tan (90° – A) = cot A
- cot (90° – A) = tan A
- sec (90° – A) = cosec A
- cosec (90° – A) = sec A
3. How are the sine and cosine of complementary angles related?
The sine of an angle and the cosine of its complement are equal. This relationship is a direct result of the definitions of sine (opposite/hypotenuse) and cosine (adjacent/hypotenuse) in a right-angled triangle. If A and B are complementary angles, then sin(A) = cos(B) and cos(A) = sin(B). This is one of the most frequently used complementary angle identities.
4. Can you show an example of simplifying a trigonometric expression using complementary angles?
Certainly. Consider the problem of evaluating tan(35°) / cot(55°). To solve this, we use the complementary angle identity for cotangent.
- Step 1: Recognise that 35° + 55° = 90°, so the angles are complementary.
- Step 2: Rewrite cot(55°) using the identity cot(A) = tan(90° - A). Here, cot(55°) = tan(90° - 55°) = tan(35°).
- Step 3: Substitute this back into the original expression: tan(35°) / tan(35°).
- Step 4: The result is 1. This shows how these identities simplify expressions without needing to find the actual angle values.
5. Why does sin(90° − A) equal cos(A) in a right-angled triangle?
This identity comes directly from the geometry of a right-angled triangle. Let the angles be A, B, and 90°. We know A + B = 90°, which means B = 90° - A. Let the sides opposite to angles A, B, and the right angle be 'a', 'b', and 'h' (hypotenuse) respectively.
- sin(A) is defined as (Opposite/Hypotenuse) = a/h.
- cos(B) is defined as (Adjacent/Hypotenuse). The side adjacent to angle B is 'a', so cos(B) = a/h.
Since B = 90° - A, we can substitute it into the second equation: cos(90° - A) = a/h. As both sin(A) and cos(90° - A) equal a/h, they must be equal to each other.
6. How do identities for complementary angles differ from those for supplementary angles?
The primary difference lies in the sum of the angles and the resulting relationships. Complementary angles add up to 90° and their identities relate a trigonometric function to its co-function (e.g., sin to cos). In contrast, supplementary angles add up to 180° and their identities (e.g., sin(180° - A) = sin A) relate a function to itself, but may involve a change in sign depending on the quadrant.
7. What are some practical applications of using complementary angle identities in Maths?
Beyond simplifying expressions, these identities have important applications:
- Solving Equations: They are used to solve trigonometric equations where different ratios are involved, by converting them to a common ratio.
- Calculus: They help in evaluating certain integrals and derivatives of trigonometric functions.
- Physics and Engineering: In fields like mechanics and electronics, they are used to analyse vectors and wave forms by resolving components along perpendicular axes.
- Geometry: They can be used to prove geometric properties without directly calculating angle measures.
8. For the CBSE 2025-26 session, is the topic of trigonometric ratios of complementary angles in the Class 10 Maths syllabus?
Yes. As per the official CBSE syllabus for the 2025-26 academic year, the topic 'Trigonometric Ratios of Complementary Angles' is an integral part of Unit V: Trigonometry, under the chapter 'Introduction to Trigonometry'. Students are expected to understand and apply these identities to solve problems.
9. What is a common misconception when working with secant and cosecant complementary identities?
A common mistake is incorrectly pairing the functions. Students sometimes forget that secant and cosecant are co-functions of each other. The correct identities are sec(90° – A) = cosec(A) and cosec(90° – A) = sec(A). A helpful tip is to remember the 'co-' prefix: sine pairs with co-sine, tangent pairs with co-tangent, and secant pairs with co-secant. This simple rule helps avoid confusion.
