# Trigonometric Identities and equations

• wei1006
In summary, the given trigonometric equation needs to be solved for the domain of 0°<x<360°. The relevant equations include secx=1/cosx, cosec x=1/sinx, cot x= 1/tanx, tan x=sinx/cosx, cot x=cosx/sinx, sin^2 x + cos^2 x=1, 1 + tan^2 x= sec^2 x, and 1+ cot^2 x = cosec^2 x. The solution involves rewriting the equation and solving for x.
wei1006
1) Problem statement:
Solve the trigonometric equation for the domain is. 0°<x<360°
5(sinx - cosx) = 4sinx - 3cosx

2) Relevent equations:
secx=1/cosx
cosec x=1/sinx
cot x= 1/tanx
tan x=sinx/cosx
cot x=cosx/sinx
sin^2 x + cos^2 x=1
1 + tan^2 x= sec^2 x
1+ cot^2 x = cosec^2 x

New domain:
Basic angle=
Solve for x
3)Attempt the question
5(sin x - cosx)=4sinx - 3cosx
5sinx-4sinx-5cosx+3cosx=0
sinx - 2 cos x=0

wei1006 said:
sinx - 2 cos x=0

Fine up to here, so what is the solution to this equation?

I am not sure how to continue...

How about trying to rewrite it by placing the 2 cos x on the other side and then dividing by something that makes it easier?

Thank you for your help, have solved it

## 1. What are the basic trigonometric identities?

The basic trigonometric identities include the Pythagorean identities, reciprocal identities, quotient identities, and negative angle identities. The Pythagorean identities are sin²θ + cos²θ = 1 and tan²θ + 1 = sec²θ. The reciprocal identities are cscθ = 1/sinθ and cotθ = 1/tanθ. The quotient identities are tanθ = sinθ/cosθ and cotθ = cosθ/sinθ. The negative angle identities are sin(-θ) = -sinθ, cos(-θ) = cosθ, and tan(-θ) = -tanθ.

## 2. How are trigonometric identities used to solve equations?

Trigonometric identities can be used to simplify and manipulate trigonometric expressions, making it easier to solve equations involving trigonometric functions. By substituting equivalent expressions using trigonometric identities, the equation can be transformed into a simpler form that is easier to solve.

## 3. What is the difference between an identity and an equation in trigonometry?

An identity is a statement that is true for all values of the variable, while an equation is a statement that is only true for specific values of the variable. In trigonometry, identities are used to simplify expressions, while equations are used to solve for specific values of the variable.

## 4. How can I prove a trigonometric identity?

To prove a trigonometric identity, you must manipulate one side of the equation using algebraic operations and trigonometric identities until it is equivalent to the other side of the equation. You can also use geometric proofs or verify the identity using a calculator for specific values of the variable.

## 5. What are the common mistakes when working with trigonometric identities?

Some common mistakes when working with trigonometric identities include forgetting to distribute negatives, using incorrect identities, and making algebraic errors. It is important to carefully check each step and use correct identities to avoid these mistakes. It is also helpful to practice and familiarize yourself with the common identities used in trigonometry.

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