Formula Proof and Solved Examples of Complementary Angle Identities
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 Explained Clearly
1. What are trigonometric ratios of complementary angles?
The trigonometric ratios of complementary angles state that the sine of an angle equals the cosine of its complement and vice versa. If two angles add up to 90°, they are complementary, and the identities are:
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
- cot(90° − θ) = tan θ
- sec(90° − θ) = cosec θ
- cosec(90° − θ) = sec θ
2. What is the formula for sin(90° − θ)?
The formula for sine of a complementary angle is sin(90° − θ) = cos θ. This means the sine of an angle’s complement equals the cosine of the angle itself. For example:
- If θ = 30°, then sin(60°) = cos(30°)
- Since cos(30°) = √3/2, sin(60°) = √3/2
3. Why is cos(90° − θ) equal to sin θ?
Cos(90° − θ) equals sin θ because in a right triangle, the sides opposite and adjacent switch roles for complementary angles. Mathematically, cos(90° − θ) = sin θ. In a right triangle:
- sin θ = Opposite/Hypotenuse
- cos(90° − θ) = Opposite/Hypotenuse (same side)
4. What is tan(90° − θ) equal to?
The tangent of a complementary angle is equal to the cotangent of the angle, so tan(90° − θ) = cot θ. Since:
- tan θ = Opposite/Adjacent
- cot θ = Adjacent/Opposite
5. How do you prove trigonometric identities of complementary angles?
Trigonometric identities of complementary angles are proved using a right-angled triangle. Steps to prove sin(90° − θ) = cos θ:
- Consider a right triangle with angles θ and (90° − θ).
- For angle θ: sin θ = Opposite/Hypotenuse.
- For angle (90° − θ): the opposite side becomes adjacent to θ.
- Thus, sin(90° − θ) = Adjacent/Hypotenuse = cos θ.
6. What are the six complementary trigonometric identities?
The six complementary trigonometric identities relate functions of (90° − θ) to basic trigonometric ratios. They are:
- sin(90° − θ) = cos θ
- cos(90° − θ) = sin θ
- tan(90° − θ) = cot θ
- cot(90° − θ) = tan θ
- sec(90° − θ) = cosec θ
- cosec(90° − θ) = sec θ
7. Can you give an example of complementary trigonometric ratios?
An example of complementary trigonometric ratios is sin(30°) and cos(60°), which are equal. Since 30° and 60° are complementary (30° + 60° = 90°):
- sin(30°) = 1/2
- cos(60°) = 1/2
8. How are complementary angles related in a right triangle?
In a right triangle, the two acute angles are always complementary and add up to 90°. Because of this:
- The side opposite one angle is adjacent to the other.
- Their sine and cosine values are interchanged.
9. What is the difference between complementary and supplementary angles in trigonometry?
Complementary angles add up to 90°, while supplementary angles add up to 180°. In trigonometry:
- Complementary angles use identities like sin(90° − θ) = cos θ.
- Supplementary angles use identities like sin(180° − θ) = sin θ.
10. Where are trigonometric ratios of complementary angles used?
Trigonometric ratios of complementary angles are used to simplify expressions, solve identities, and evaluate angles quickly. Common uses include:
- Simplifying trigonometric equations
- Proving trigonometric identities
- Solving right triangle problems
- Competitive exam and board exam questions
