Trigonometric Equations And Identities Examples

A trigonometric equation is a mathematical statement that relates the lengths of two sides of a right triangle to the measure of its third side. There are many different types of trigonometric equations, but all of them can be solved using the appropriate trigonometric identity or formula. In this article, we will take a look at some examples of trigonometric equations and how to solve them. We will also discuss some common mistakes that students make when solving these problems. Let’s get started!

What are Trigonometric Identities?

A trigonometric identity is an equation that’s true for all values of the variables involved. In other words, you can plug in any value you want for the variables and the equation will still be true. For example, the Pythagorean theorem is a trigonometric identity: it’s always true that a2 + b2 = c2.

There are an infinite number of trigonometric identities, but there are a few that are so important that you really need to memorize them. Below, we list some of the most important trigonometric identities and tangent formulas.

Types of Trigonometry Identities

There are three types of trigonometry identities:

Conditional Identities:

An identity is conditional if it is only true for certain values of the variables. For example, the tangent function is undefined when the angle is equal to 90 degrees or 270 degrees. So, the following identity would be classified as a conditional identity:

tan(90°) = 0. This identity is only true when the angle is equal to 90 degrees.

Pythagorean Identities:

A Pythagorean identity is an equation that is true for right-angled triangles. For example, in a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. This can be represented using trigonometric functions as:

sin22 + cos22 =  csc22 – cot22.

This identity is true for all values of the angle, not just right-angled triangles.

Reciprocal Identities:

A reciprocal identity is one where two sides of an equation are reciprocals of each other. For example:

sin(x) =  1/csc(x). This identity is true for all values of x.

What is Tangent in Trigonometry?

The tangent of an angle is the ratio of the length of the side opposite to the angle to the length of the adjacent side. It is represented by the symbol tan. Tangent is one of the basic functions of trigonometry.

The tangent function has several important properties, which are used in many formulae.

Some of these properties are:

– The tangent function is odd, that is, tangent(-x)=-tan(x).

– The tangent function is periodic, that is, tangent(x+y)=tan(x)tan(y).

– The tangent function is undefined at x= (n+½)π, where n is any integer.

Some of the important formulae involving tangent are:

– tangent(x)=sin(x)/cos(x)

– tangent(x)=cot(x)*tan(x)

– tangent(x+y)= (tan(x)+tan(y))/(tan(x)*tan(y))

– tangent(x-y)= (tan(x)-tan(y))/(tan(x)*tan(y))

Trigonometry Identities and equations.

Trigonometry identities are very important in solving mathematical problems. The tangent function is one of the most important functions in trigonometry. Some of the common trigonometry equations are :

1- sin2x = cos2x

1- tan2x = sec2x

1- cot2x = -cosec2x

Some of the trigonometric formulae are:

sin(A+B) = sinAcosB + cosAsinB

cos(A+B) = cosAcosB – sinAsinB

– tan(A+B) = tanAtanB / (tanA + tanB)

– cot(A+B) = cotAcotB / (cotA + cotB)

– sin(A-B) = sinAcosB – cosAsinB 

– cos(A-B) = cosAcosB + sinAsinB

– tan(A-B) = (tanA – tanB) / (tanAtanB)

Trigonometry Problems Based on Trigonometry Identities

Some of the solved examples of trigonometry identities are given below:

Example 1:  Prove the identity: tangent²θ – secant²θ = -cotan²θ

Solution: Given identity is tangent²θ – secant²θ = -cotan²θ

We know that, tangent θ = sin θ/cos θ

secantθ =  cos θ/sin θ

cotangentθ = cos θ/sin θ

Now, tangent²θ = (sin θ/cos θ)²

= (sinθ)²/(cosθ)²

tangent²θ – secant²θ = (sinθ/cosθ)² – (cosθ/sinθ)²

= (sinθ/cosθ)(sinθ/cosθ) – (cosθ/sinθ)(cosθ/sinθ)

= (sin²θ/cos²θ) – (cos²θ/sin²θ)

= (-cotan²θ)

Hence, Proved.

Example: Prove the identity: cosecant²θ – cotangent²θ = cosecant²θ

Solution: Given identity is cosecant²θ – cotangent²θ = cosecant²θ

We know that, cosecant θ =

Now, cosecant²θ = (sin θ/cos θ)²

= (sinθ)²/(cosθ)²

cosecant²θ – cotangent²θ = (sinθ/cosθ)² – (cosθ/sinθ)²

= (sinθ/cosθ)(sinθ/cosθ) – (cosθ/sinθ)(cosθ/sinθ)

= (sin²θ/cos²θ) – (cos²θ/sin²θ)

= (-cotan²θ)

Hence, Proved.

Conclusion

In conclusion, these examples demonstrate the various trigonometric identities and how to solve equations that involve them. tangent, formulae, and trigonometry problems are all important concepts to understand in order to be successful in solving these types of equations. Trigonometric equations and identities mean a lot in mathematics. These examples help to provide a better understanding of how to work with these equations and identities. In this article, we learnt about trigonometric equations and solved some trigonometry problems. With this knowledge, you can go on to solve more complex equations and problems. Thanks for reading! I hope this was helpful! Please comment below if you have any questions.