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We can define a triangle as a geometric figure that has three sides. There are a variety of triangle’s shapes and sizes. One of the three sides of a right angle triangle is a right angle, or 90 degrees. The base is the side that is horizontal to the plane and adjacent to the right angle. The perpendicular is the side of the triangle that is 90⁰ degrees from the base. The sum of the other two angles other than the right angle in each right-angle triangle is 90⁰. The Pythagorean theorem asserts that “the square on the hypotenuse is equal to the sum of the squares on the other two sides” for all right angle triangles.
Trigonometric ratios
Trigonometry is a branch of mathematics that deals with the relationship between sides and angles of triangles, particularly right triangles. There are six trigonometric ratios in terms of sides that are used to depict the angles of a right triangle . There are two angles in every right triangle that are not 90⁰ degrees.
If any angle other than a right angle is designated as angle ‘A’ in a right triangle, the side next to angle ‘A’ that is not a hypotenuse is referred to as the adjacent side or base, and the side opposite to angle ‘A’ is referred to as the opposite side or perpendicular. The all the trigonometric ratios are given below :-
SinA = P / H
CosA = B / H
TanA = P / B
CosecA = H / P
SecA = H / B
CotA = B / P
Where P , B and H are the perpendicular , base and height of the right-angled triangle.
The sine and cosine of any angle can be deduced from the previous definitions as the two fundamental trigonometric ratios from which all other trigonometric ratios can be defined. The ratio of an angle’s sine to its cosine is its tangent. The multiplicative inverse of an angle’s sine is its cosecant. The multiplicative inverse of an angle’s cosine is its secant. The multiplicative inverse of an angle’s tangent is its cotangent. The ratio of an angle’s cosine to its sine is also known as the cotangent.
Proof of trigonometric ratios of complementary angles
If the total of two angles equals 90⁰ , they are said to be complementary. The value obtained by subtracting any angle from 90⁰ is the complement of that angle. The total of the other two angles in a right triangle, except the right angle, equals 90⁰ . As a result, these two angles are seen to be complimentary.
Consider a right triangle ABC with a right angle at B to calculate the complementary angles formula’s trigonometric ratios. When the angle at “ A ” is used as the reference angle, the other reference angle at “ C ” becomes the complement of the angle at “ A .” i.e. angle at point ‘ C ‘ = 90⁰ – A . The opposing side is ‘BC’ and the adjacent side is AB when A is used as the reference angle. AC is the hypotenuse of a right triangle because it is opposite the right angle. The reference angle” trigonometric ratios are as follows:
SinA = P / H
CosA = B / H
TanA = P / B
CosecA = H / P
SecA = H / B
CotA = B / P
And when we take a reference of angle C which is complement of A then the perpendicular and base interchanges then
SinC = B / H = Sin ( 90° – A ) = CosA
CosC = P / H = Cos ( 90° – A ) = SinA
TanC = B / P = Tan ( 90° – A ) = CotA
CosecC = H / B = Cosec ( 90° – A ) = SecA
SecC = H / P = Sec ( 90° – A ) = CosecA
CotC = P / B = Cot ( 90°- A ) = TanA
From the above observation we can conclude the following results :
Sin ( 90° – A ) = CosA
Cos ( 90° – A ) = SinA
Tan ( 90° – A ) = CotA
Cosec ( 90° – A ) = SecA
Sec ( 90° – A ) = CosecA
Cot ( 90°- A ) = TanA
Conclusion :-
It’s all about the complementary angles’ trigonometric ratios. The complementary angle’s trigonometric function is defined as another trigonometric function of the original angle, according to the trigonometric complementary ratio theorem. Because the complement of 45⁰ is equally equal to 45⁰ , trigonometric ratios of 45⁰ and its complement are always the same.
