Relationships between Trigonometric Functions (Trigonometric Identities)

WHAT ARE RELATIONSHIPS BETWEEN TRIGONOMETRIC FUNCTIONS?

Relationships between trigonometric functions, also known as trigonometric identities, are fundamental building blocks of mathematics, especially in trigonometry and circular geometry. These functions include sine (sin), cosine (cos), and tangent (tan), along with their reciprocal values: cosecant (csc), secant (sec), and cotangent (cot). These functions are interconnected through numerous mathematical relationships (identities) that play an important role in deriving trigonometric equations, understanding geometric relationships, and solving complex problems.

BASIC RATIOS BETWEEN TRIGONOMETRIC FUNCTIONS

The basis for understanding trigonometric functions comes from the right-angled triangle. In such a triangle, the definition of each trigonometric function can be expressed as a ratio between the lengths of its sides.

• SINE (sin): is the ratio of the length of the opposite side to the length of the hypotenuse.
• COSINE (cos): is the ratio of the length of the adjacent side to the length of the hypotenuse.
• TANGENT (tan): is the ratio of the length of the opposite side to the length of the adjacent side, which can also be expressed as sin/cos.
• COTANGENT (cot): is the ratio of the length of the adjacent side to the length of the opposite side, which can also be expressed as cos/sin.

FUNDAMENTAL IDENTITIES

These are fundamental for connecting sine and cosine, as well as tangent and cotangent. Their common feature is that they can be derived from the Pythagorean theorem, which applies in a right-angled triangle, so they can also be called Pythagorean identities:

• sin²x + cos²x = 1
• 1 + tan²x = sec²x (which is 1/cos²x)
• 1 + cot²x = csc²x (which is 1/sin²x)

And the quotient identities:

• tan x = sin x / cos x
• cot x = cos x / sin x

RELATIONSHIPS BETWEEN FUNCTIONS (ADVANCED IDENTITIES)

Trigonometric functions are also connected through various other relationships and transformations:

• SUM AND DIFFERENCE FORMULAS (ADDITION THEOREMS): These formulas express sin(x ± y), cos(x ± y), and tan(x ± y) in terms of the sines, cosines, and tangents of angles x and y.
• DOUBLE-ANGLE AND HALF-ANGLE FORMULAS: These formulas relate the functions of a double angle (e.g., sin(2x)) or a half angle (e.g., sin(x/2)) to the original functions.

USES

Relationships between trigonometric functions (identities) are particularly used for simplifying various mathematical expressions where a simpler result is desired. They are also encountered in trigonometry when finding the values of sine, cosine, tangent, or cotangent of an angle x, and are likewise used in solving trigonometric equations.

CONCLUSION

The relationships between trigonometric functions (trigonometric identities) are the core of trigonometry, as they allow for the conversion and interconnection of different trigonometric functions. Their utility ranges from basic equation solving to advanced mathematical analysis. Understanding these relationships not only facilitates solving trigonometric problems but also opens the door to the world of more complex mathematical areas, such as the analysis of waves, harmonic motions, and the study of periodic phenomena in physics and engineering.