A Short note on the Identities in Trigonometry

Sin, cos, and tan are the three main trigonometric ratios. The reciprocals of sin, cos, and tan are the three different trigonometric ratios in trigonometry: sec, cosec, and cot. What is the relationship between these trigonometric ratios (sin, cos, tan, sec, cosec, and cot)? Trigonometric identities bind them together (or, in short, trig identities). Before understanding trigonometric formulas, trigonometry formula, trigonometric formula, you need to be comfortable with algebra and geometry. You should be able to manipulate algebraic expressions and solve problems after learning algebra. You should know about similar triangles, the Pythagorean theorem, and a few other things from geometry, but not a lot.

What is Trignometry?

Trigonometry is a discipline of mathematics concerned with the study of angles, angle measurement, and measurement units. It also looks at the six ratios for a particular angle and the relationships that these ratios satisfy. The study is also of the angles that make up the triangle’s constituents in a broader sense. A description of a triangle’s characteristics, solving a triangle; and physical difficulties in the field of heights and distances utilizing the properties of a triangle are all part of the study. It also includes a technique for solving trigonometric equations.

Trigonometric Identities:

If an equation using trigonometric ratios of an angle is valid for all values of the angle, it is termed trigonometric Identity. These are beneficial when trigonometric functions are used in an expression or an equation. Sine, cosine, tangent, cosecant, secant, and cotangent are the six basic trigonometric ratios. The sides of the right triangle, such as the adjacent, opposing, and hypotenuse sides, are used to determine these trigonometric ratios and trigonometric formula

For example, some of the algebraic identities are:

• (a + b)² = a² + 2ab + b²

• (a – b)² = a² – 2ab+ b²

• (a + b)(a-b)= a² – b²

The algebraic identities are only concerned with the variables, but the trig identities are the six trigonometric formulas sine, cosine, tangent, cosecant, secant, and cotangent.

Complementary and Supplementary Trigonometric Identities

The trigonometric ratios of complementary angles are:

• sin (90°- θ) = cos θ

• cos (90°- θ) = sin θ

• cosec (90°- θ) = sec θ

• sec (90°- θ) = cosec θ

• tan (90°- θ) = cot θ

• cot (90°- θ) = tan θ

The trigonometric ratios of supplementary angles are:

• sin (180°- θ) = sinθ

• cos (180°- θ) = -cos θ

• cosec (180°- θ) = cosec θ

• sec (180°- θ)= -sec θ

• tan (180°- θ) = -tan θ

• cot (180°- θ) = -cot θ

Reciprocal Trigonometric Identities

• sin θ = 1/cosecθ (OR) cosec θ = 1/sinθ

• cos θ = 1/secθ (OR) sec θ = 1/cosθ

• tan θ = 1/cotθ (OR) cot θ = 1/tanθ

Pythagorean Trigonometric Identities

• sin²θ + cos²θ = 1

• 1 + tan²θ = sec²θ

• 1 + cot²θ = cosec²θ

Sum and Difference Trigonometric Identites

• sin (A+B) = sin A cos B + cos A sin B

• sin (A-B) = sin A cos B – cos A sin B

• cos (A-B) = cos A cos B + sin A sin B

• tan (A+B) = (tan A + tan B)/(1 – tan A tan B)tan (A-B) = (tan A – tan B)/(1 + tan A tan B)

Sine and Cosine Rule Trigonometric Identities

The sine rule establishes the relationship between a triangle’s angles and its sides. We’ll have to employ the sine and cosine rules for non-right-angled triangles. The sine rule may be written as follows for a triangle with sides ‘a’, ‘b’, and ‘c’, and opposing angles A, B, and C:

• a/sinA = b/sinB = c/sinC

• sinA/a = sinB/b = sinC/c

• a/b = sinA/sinB; a/c = sinA/sinC; b/c = sinB/sinC

When two sides and the included angle of a triangle are supplied, the cosine rule is used to determine the relationship between the angles and the sides of the triangle. For a triangle with sides a, b, and c and opposing angles A, B, and C, the sine rule is

• a² = b² + c² – 2bc·cosA

• b² = c² + a² – 2ca·cosB

• c² = a² + b² – 2ab·cosC

Conclusion:

Trigonometry is used to set directions such as north, south, east, and west, and it informs you of the compass direction to follow to travel on a straight path. It’s used in navigation to find a certain spot. It’s also used to calculate the distance between the coast and a location in the water. The angles formed by the ratios of trigonometric functions are known as trigonometry angles. The study of the connection between angles and triangle sides is known as trigonometry. and I have also shared the study about trigonometric formulas, trigonometric formulas, and trigonometry formulas.  Without an understanding of trigonometry, it will be difficult to build buildings, automobiles, and other structures.