Basic Identities of Trigonometric Functions

Trigonometry is the discipline of mathematics that studies the connection between the sides and angles of a right-angled triangle. Trigonometric ratios, such as sine, cosine, tangent, cotangent, secant, and cosecant, are employed to examine this connection. The notion of trigonometry was established by the Greek mathematician Hipparchus, while the name trigonometry is a 16th century Latin derivation. One of the most significant fields of mathematics is trigonometry. The terms ‘Trigonon’ and ‘Metron,’ which imply triangle and measure respectively, are combined to produce the word trigonometry. The link between the sides and angles of a right-angled triangle is studied. Using equations and identities based on this relationship, it is easier to compute the measure of unknown dimensions of a right-angled triangle.

The measurement of angles and issues involving angles are covered in trigonometry fundamentals. Trigonometry has some basic functions: sine, cosine, and tangent. Other essential trigonometric functions may be derived from these three fundamental ratios or functions: cotangent, secant, and cosecant.

Ratios of Trigonometry

In trigonometry, there are six essential ratios that assist in creating a link between the ratio of sides of a right triangle and the angle. If is the angle created by the base and hypotenuse of a right-angled triangle, then, (Here Z is the angle between hypotenuse and base.)

• sin Z = Perpendicular/Hypotenuse

• cos Z= Base/Hypotenuse

• tan Z= Perpendicular/Base

The values of the other three functions are cosec, sec, and  cot, and are determined by the values of tan, cos, and sin, as shown below.

• cot Z= 1/tan Z= Base/Perpendicular

• sec Z= 1/cos Z= Hypotenuse/Base

• cosec Z= 1/sin Z= Hypotenuse/Perpendicular

Trigonometric Identities

Trigonometric identities can be defined as the equations that shows to various trigonometric functions and are true for all values of the variable in the domain. A mathematical identity is a formula that holds true for all values of the variables in it.

For instance, here are some algebraic identities: 

• (P + W)2 = P2 + 2PW + W2
• (P – W)2 = P2 – 2PW+ W2
• (P + W) (P-W) = P2 – W2

The algebraic identities connect only the variables, whereas the trig identities connect the six trigonometric functions sine, cosine, tangent, cosecant, secant, and cotangent. Let’s take a closer look at each sort of trigonometric identity.

Reciprocal Identities in Trigonometry

The  cosecant, secant, and cotangent are the reciprocals of sin, cosine, and tangent, respectively.

Thus, the reciprocal identities are as follows:

• sin Z = 1/cosec Z (OR) cosec Z = 1/sin Z 

• cos Z = 1/sec Z (OR) sec Z = 1/cos Z 

• tan Z = 1/cot Z (OR) cot Z = 1/tan Z 

Trigonometric Pythagorean Identities

Pythagoras’ theorem in trigonometry is the source of the Pythagorean trigonometric identities. When the Pythagoras theorem is applied to the right-angled triangle below, the following results are obtained:

Opposite2 + Adjacent2 = Hypotenuse2

Dividing both sides by Hypotenuse2

Opposite2/Hypotenuse2 + Adjacent2/Hypotenuse2 = Hypotenuse2/Hypotenuse2

sin2Z  + cos2Z = 1

One of the Pythagorean identities is this. We may derive two more Pythagorean trigonometric identities in the same way.

• 1 + tan 2Z = sec 2Z
• 1 + cot 2Z = cosec 2x

Trigonometric Identities that are Complementary and Supplementary

Complementary angles are a pair of two angles whose total equals 90°. (90° -Z) is the required complement of an angle. Complementary angle trigonometric ratios are :

• sin (90°- Z) = cos Z

• cos (90°- Z) = sin Z

• cosec (90°- Z) = sec x

• sec (90°- Z) = cosec Z

• tan (90°- Z) = cot Z

• cot (90°- Z) = tan Z

The supplementary angles are a pair of 2 angles and the sum of these angles are  equal to 180°. An angle’s complement is (180° -Z). Supplementary angle trigonometric ratios are:

• sin (180°- Z) = sin Z

• cos (180°- Z) = -cos Z

• cosec (180°- Z) = cosec Z

• sec (180°- Z)= -sec Z

• tan (180°- Z) = -tan Z

• cot (180°- Z) = -cot Z

Difference and Sum Identities Trigonometric

The formulae sin(P+R), cos( P-R), cot ( P+R), and others are part of the sum and difference identities.(Here, P and R are angle)

• sin (P + R) = sin P cos R+ cos P sin R

• sin (P -B) = sin P  cos R – cos P  sin R

• cos (P +B) = cos P  cos R – sin P  sin R

• cos (P -B) = cos P  cos B + sin P  sin R

• tan (P +B) = (tan P  + tan R)/(1 – tan P  tan R)

• tan (P -B) = (tan P  – tan R)/(1 + tan P  tan R)

Double angle formulae

 The sum and difference formulas may be used to find the double angle trigonometric identities.

For instance, using the formulae above:

sin (P + R) = sin P cos R+ cos P sin R

Substituting P = R = Z on both sides here, we get:

sin (Z  + Z ) = sinZ  cosZ  + cosZ  sinZ 

sin 2Z  = 2 sinZ  cosZ 

We can derive the other double-angle identities in the same way.

sin 2Z  = 2 sinZ  cosZ 

cos 2Z  = cos2Z  – sin 2Z 

= 2 cos2Z  – 1

= 1 – sin 2 Z 

tan 2Z  = (2tanZ ) / (1 – tan2Z )

Conclusion

Many identities, or equations that are true for all conceivable inputs, have been observed in trigonometry. Trigonometric identities are identities that simply include angles. Triangle identities are equations that link both the sides and angles of a triangle.