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Trigonometry Identities

19 Oct 2015Homework Help Online

In mathematics identities hold the quality of always being true.

For instance: if x/10 = x*2

An identity promises to be true, no matter what values are put in the place of x to prove the identity.

For instance:  X/10 = X*2

Whatever, values can be put in the place of x to prove the true nature of the identity.

Trigonometric identities prove to be useful in simplifying complex equations and expressions into proper forms. They may appear hard nuts to crack at first sight, but right application of identities and modulation of formulas make it a game to play. In this guide, let's teach you the rules of “Trig game”.

Trigonometry talks all about triangles and Right Angled Triangles where it strikes the most.

Source: Wiki

"Sine, Cosine and Tangent" are the major keys of trigonometry that help unlock the missing side or angle in a triangle.

These three are literally one side of a right-angled triangle divided by another.

For any angle "θ":

Sine Function:

sin(θ) = Opposite / Hypotenuse

Cosine Function:

cos(θ) = Adjacent / Hypotenuse

Tangent Function:

tan(θ) = Opposite / Adjacent

(We often abbreviate sine, cosine and tangent as sin, cos and tan)

There are endless trigonometry identities, but following are used abundantly.

Pythagorean Theorem: In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

It is stated in this formula:

a2 + b2 = c2

Reciprocal identities: In mathematics, reciprocal of a number is 1 divided by that number. For example, the reciprocal of 6 is 1/6. In trigonometry, reciprocal of a trig function is 1.  

Angle-Sum and -Difference Identities: The angle sum and difference formulas for sine and cosine are occasionally known as Simpson's formulas. To find the sum or difference of two angles following trigonometry formulas can be applied.

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)

sin(α – β) = sin(α)cos(β) – cos(α)sin(β)

cos(α + β) = cos(α)cos(β) – sin(α)sin(β)

cos(α – β) = cos(α)cos(β) + sin(α)sin(β)

Double-Angle Identities: Double-angle formulas can be elaborated to multiple-angle functions by using the formulas systematically.

sin(2x) = 2sin(x)cos(x)

cos(2x) = cos2(x) – sin2(x) = 1 – 2sin2(x) = 2cos2(x) – 1

Co-Function Identities: The sine is the co function of cosine.

Sum-to-Product Formulas: This formula is applicable whenever the need arises to convert the sum of sine and cosine into a product.

Product-to-Sum Formulas: This formula is applicable whenever need rises to convert the product of sine and cosine into a line of stitching.

Power-Reducing/Half Angle Formulas:  Power reduction formulas, double angle and half angle formulas work in the almost similar manner. Power reduction formulas can be easily derived through the use of double-angle and half-angle formulas and the Pythagorean identity).

The use of a power reduction formula reveals the quantity without the exponent.

Even-Odd Identities in Trigonometric Functions

Knowledge of expression's even or odd nature helps to solve the problem easily. These even-odd identities work where the variable of an expression inside the trig function is negative (such as –x). The even-odd identities are as follows:

sin(–x) = –sinx

csc(–x) = –cscx

cos(–x) = cosx

sec(–x) = secx

tan(–x) = –tanx

cot(–x) = –cotx

Eventually, the daunting looking trigonometry equations have started appearing as an interesting game. These trig characters may seem to be confusing enough but once clear with their concept it’s as easy as ABC.

A few links to help you understand the Trigonometry Identities:

  1. http://www.sosmath.com/trig/Trig5/trig5/trig5.html This link can help you understand the identities in detail with equations.
  2. https://www.mathsisfun.com/algebra/trigonometric-identities.html The site assists with the interesting and fun loving ways to learn Trigonometry Identities.
  3. https://www.khanacademy.org/math/trigonometry/less-basic-trigonometry The site can teach you all about the trig identities and will give you the detailed theoretical knowledge of the topic.

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