Trigonometric Identities

TRIGONOMETRIC IDENTITIES

Trigonometric identities serve as essential mathematical tools that establish connections between different trigonometric functions. These relationships are universally valid, holding true for all values within the specified domains. Trigonometric identities play a pivotal role in simplifying complex trigonometric expressions, proving mathematical assertions, and solving equations involving trigonometric functions.

DOUBLE ANGLE TRIGONOMETRIC IDENTITIES

RECIPROCAL IDENTITIES

Reciprocal identities expresses a trigonometric function in a reciprocal form. Reciprocal identities establish a relationship between three trigonometric functions sine, cosine, tangent and their reciprocals cosecant, secant, tangent.

1. Cosecant Reciprocal Identities

$csc \theta =\frac{1}{sin \theta}$

2. Secant Reciprocal Identities

$sec \theta =\frac{1}{cos \theta}$

3. Cotangent Reciprocal Identities

$cot \theta =\frac{1}{tan \theta}$

QUOTITENT IDENTITIES

Quotient identities expresses a one trigonometric function as a quotient of two others. There are two primary quotient identities

1. Quotient identity for tangent

$tan \theta =\frac{sin\theta}{cos\theta}$

This identity is very useful to make a connection with the three major trigonometric functions sine, cosine, tangent

2. Reciprocal identity for cotangent

$cot \theta =\frac{cos\theta}{sin\theta}$

This identity is very useful to make a connection with the three minor trigonometric functions sine, cosine, tangent

PYTHAGOREAN IDENTITIES

Pythagorean identities are set of three trigonometric identities that are derived from the Pythagorean theorem. These identities involve 3 major trigonometric functions sine, cosine and tangent.

1. Sine Squared Identity $sin^{2}\theta+cos^{2}\theta=1$

sin^{2}\theta=1-cos^{2}\theta

$cos^{2}\theta=1-sin^{2}\theta$

2. Secant Squared Identity $sec^{2}\theta=1+tan^{2}\theta$

$sec^{2}\theta-tan^{2}\theta=1$

$tan^{2}\theta=sec^{2}\theta-1$

3. Cosecant Squared Identity $csc^{2}\theta=1+cot^{2}\theta$

$csc^{2}\theta-cot^{2}\theta=1$

$cot^{2}\theta=csc^{2}\theta-1$

SUM AND DIFFERENCE TRIGONOMETRIC IDENTITIES

The sum and difference trigonometric identities are formulas that express two angles as a form of sum or difference.

There are six sum and difference formulas. We can use the sum and difference formulas for Sine, Cosine and Tangent functions.

These trigonometric identities are used to find some unknown angles and simplifying expressions and solving trigonometric equations.

Here are the list of sum and difference formula

sin (A + B) = sinA cosB + cosA sinB
sin (A – B) = sinA cosB – cosA sinB
cos (A + B) = cosA cosB – sinA sinB
cos (A – B) = cosA cosB + sinA sinB
tan(A+B) = $\frac{tanA+tanB}{1-tanAtanB}$
tan(A-B) = $\frac{tanA-tanB}{1+tanAtanB}$

EVEN TRIGONOMETRIC IDENTITIES

Even trigonometric functions satisfy the following condition f(x) = f(-x) for all x values. Even function applied to angle and its negative angle will give you the same results.

Major trigonometric functions even identities

\sin (-\theta) =-sin\theta

\cos (-\theta) =cos\theta

\tan (-\theta) =-tan\theta

Minor trigonometric functions even identities

\csc (-\theta) =-csc\theta

\sec (-\theta) =sec\theta

\cot (-\theta) =-cot\theta

Even identities are useful in simplifying expressions and providing trigonometric identities relationships and solving trigonometric equations.

DOUBLE ANGLE TRIGONOMETRIC IDENTITIES

Double angle trigonometric identities express trigonometric functions of twice an angles in terms of the trigonometric functions of that angle. Here are the double angle trigonometric identities

Sine Double Angle Identity

\sin (2\theta) =2sin(\theta)cos(\theta)

Cosine Double Angle Identity

\cos (2\theta) =cos^{2}(\theta)-sin^{2}(\theta)

\cos (2\theta) =2cos^{2}(\theta)-1

\cos (2\theta) =1-2sin^{2}(\theta)

Tangent Double Angle Identity

tan2\theta = \frac{2tan\theta}{1-tan^2\theta}

We can derive these double angle identities from various trigonometric identities and formula such as the sum and difference identities and these identities are applicable for all values of $\theta$ .

There is also half angle trigonometric identities which expresses trigonometric functions in half angle in terms of the original trigonometric functions. These identities are obtained by rearranging the double angle identities.

HALF ANGLE TRIGONOMETRIC IDENTITIES

Half angle trigonometic identities are obtained by rearranging the double angle identities.

Sine half angle identity

$\sin \frac{\theta }{2}=\pm\sqrt{\frac{1-cos\theta}{2}}$

Positive or negative square root depends sign on the quadrant in which $\frac{\theta}{2}$ lies.

Cosine half angle identity

\cos \frac{\theta }{2}=\pm\sqrt{\frac{1+cos\theta}{2}}

Positive or negative square root depends sign on the quadrant in which

\frac{\theta}{2}

lies.

Tangent half angle identity

\tan \frac{\theta }{2}=\pm\sqrt{\frac{1-cos\theta}{1+cos\theta}}

Positive or negative square root depends sign on the quadrant in which

\frac{\theta}{2}

lies.

These identities are obtained by applying the double-angle trigonometric identities and solving for the half-angle expressions. They are very helpful for simplifying trigonometric expressions involving half angles.