Function of Trigonometry

All the trigonometric functions will be discussed in association with the angle R. 

∠R is situated opposite to the perpendicular PQ of the right-angled triangle PQR. 

Side PQ is the hypotenuse of the △PQR. 

Side QR is the base of the triangle. 

1.Sine: The ratio between the opposite arm of an angle in a right-angled triangle and its hypotenuse provides us with the sine of that specific angle. In the above specimen side, PQ is drawn opposite to the ∠R while PR is the hypotenuse of △PQR. 

 ∴ sin R = PQ/PR

2.Tangent: The ratio between the length of the opposite arm of the concerned angle in a right-angled triangle and its adjacent side gives us the tangent of that angle. Here, QR is the adjacent side of ∠R. Also the side PR is adjacent, but we do not consider the hypotenuse in this case. 

               ∴ tan R = PQ/QR

3.Cosine: In a right-angled triangle, the cosine function can be simply determined by finding the ratio between the length of the adjacent arm of the concerned angle and the hypotenuse. In the case of triangle PQR, cos R = QR/PR. 

Three other trigonometric functions are applied in less frequency. These are derived from the above three basic functions and each of them is the reciprocals of their parent functions. 

As they are reciprocals, we can easily deduce their value by flipping the ratio. 

Cosec R = 1 / sin R [ cosec function is the reciprocal of sin function] 

∴ cosec R = PR / PQ 

Similarly, we can deduce the sec function of ∠R from cosine R. 

Sec R = 1 / cos R

∴ sec R = PR / QR

Cotangent or cot function is the reciprocal of tangent. 

∴ cot R = 1 / tan R

or, cot R = QR / PQ

These are the trigonometric formulas for each of the angular functions. 

Trigonometric Identities 

Trigonometric identities are some key formulas that help us to solve several equations when the standard trigonometric functions are used in any expression. These identities satisfy each value of the variables associated with the trigonometric operation. The only condition is the triangle must be right-angled. Complex geometric problems are simplified to a great extent when we apply these trigonometric identities. 

We can classify trigonometric identities into eleven types:

1.Reciprocal identities 

An example of reciprocal identity: sin β = 1/ cosec β

2. Pythagorian identities 

Example : 1 + tan² R  = sec² R

3.Identities of trigonometric opposite angles

Cot (-β) = – cotβ, sin (-α) = – sin α and so on for the other trigonometric functions. 

4.Trigonometric identities involving complementary angles

Examples are: Tan (90 – R) = Cot R

Cot ( 90 – R) = Tan R

5.Ratio trigonometric identities 

Example : Tan α = sin α/ cos α 

6.Trigonometric identities involving supplementary angles

These identities associate the angles whose sum is equal to 180°. For example, sec (180° – β) = – sec β

7.Trigonometric functional identities of products

For the angles P and R of the above △PQR, sin P. sin R = [cos (P- R) – cos (P + R)] /2

8.Product-sum identities 

For the angles P and Q,

 sin P + sin Q = 2 sin (P + Q) /2. Cos (P – Q) /2

9.Double angle identities 

The equation tan 2β = (2 tan β) / (1 – tan² β) holds true if the angle β is doubled. 

10.Half angle identities 

Cos (β/2) = ±√(1 + cos β)/2 

This equation is true if the angle denoted by variable β is halved. 

11.Trigonometric identities – Sum and difference 

Tan (P + R) = (tan P + tan R) / (1 – tan R. tan P) 

Conclusion 

The modern definition of trigonometric functions does not explain them only in case acute angles. We can assess the domain of the functions on a complex plane using trigonometric operations. An alternative superscript of – 1 gives the inverse functions of standard trigonometric functions.