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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:
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Sine of an angle ‘A’ is given as
[sin A = frac{{opposite}}{{Hypotenuse}} = frac{{BC}}{{AC}}]
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Cosine of an angle ‘A’ is given as
[cos A = frac{{Adjacent}}{{Hypotenuse}} = frac{{AB}}{{AC}}]
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Tangent of an angle ‘A’ is given as
[tan A = frac{{Opposite}}{{Adjacent}} = frac{{BC}}{{AB}} = frac{{sin A}}{{cos A}}]
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Cosecant of an angle ‘A’ is given as
[cos ecA = frac{{Hypotenuse}}{{Opposite}} = frac{{AC}}{{BC}} = frac{1}{{sin A}}]
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Secant of an angle ‘A’ is written as
[sec A = frac{{Hypotenuse}}{{Adjacent}} = frac{{AC}}{{AB}} = frac{1}{{cos A}}]
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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.
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The tangent of an angle is the ratio of its sine to its cosine.
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The cosecant of an angle is the multiplicative inverse of its sine.
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The secant of an angle is the multiplicative inverse of its cosine.
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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 ectheta = 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 ecleft( {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:
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Inference |
Reason |
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[Sinleft( {{{90}^0} – A} right) = Cos{text{ }}A] |
Comparing equations (1) and (7) |
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[Cosleft( {{{90}^0} – A} right) = SinA] |
Comparing equations (2) and (8) |
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[Tanleft( {{{90}^0} – A} right) = CotA] |
Comparing equations (3) and (9) |
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[Cotleft( {{{90}^0} – A} right) = TanA] |
Comparing equations (4) and (10) |
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[Secleft( {{{90}^0} – A} right) = CosecA] |
Comparing equations (5) and (11) |
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[Cosecleft( {{{90}^0} – A} right) = SecA] |
Comparing equations (6) and (12) |
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.