Trigonometric Identities algebra

• Mspike6
Good job!In summary, solving trigonometric equations often involves using identities and factoring to simplify the equation and find solutions within a given domain. It is important to be familiar with common identities and techniques for factoring trigonometric expressions in order to solve these types of equations.
Mspike6
hello.

i have a question about trignometric identities.. it's realtivly easy, but am struggling with the algebra in it ( Algebra + trig = Very confusing to me )

Prove that ..

[Sinx/(1+Cosx)] + [(1+cosx) / sinx] = 2csc x

i manged to get it to [Sin2x+1+2cosx+cos2x] / SinxCosx

but i don't know what to do after that

any help would be appreciated, even if hints or steps without actually solving it for me .

Thank you

Mspike6 said:
hello.

i have a question about trignometric identities.. it's realtivly easy, but am struggling with the algebra in it ( Algebra + trig = Very confusing to me )

Prove that ..

[Sinx/(1+Cosx)] + [(1+cosx) / sinx] = 2csc x

i manged to get it to [Sin2x+1+2cosx+cos2x] / SinxCosx

but i don't know what to do after that

any help would be appreciated, even if hints or steps without actually solving it for me .

Thank you

There's an identity involving sin2x and cos2x that might help get you a bit farther...

For the first term sinx/(1 + cosx), multiply everything by the conjugate of the denominator. It should be a lot simpler than what you tried doing.

Mspike6 said:
i manged to get it to [Sin2x+1+2cosx+cos2x] / SinxCosx
That should actually be $$\frac{sin^{2}x+1+2cosx+cos^{2}x}{sinx+sinxcosx}$$

Simplify and then factor.

Woohoo got it ! ..

Sin2x + cos2x will be 1

so ..

it will be [2 + 2cosx] / [sinx + sinxcosx]

Cosx cancels each other

==> 4/ 2sinx

==> 2/sinx

==> 2cscx

Thanks a lot for the help guys :D

I have anther question...

Find the general solutions for the following equations.

2 cos2 x -1 =0

Soultion.

Cos2x = 1/2
Cosx= - or + Sqrt 2 / 2

The general solution is pi /4 +2nPi, 3pi/4 +2nPi , 5Pi/4 + 2nPi and 7Pi/4 +2nPi , where n is an integer

But why can't i just take the Double angle of 2cos2x ?

which will be, cos2x=0

The general solution is Pi/2 +2nPi and 3Pi/2 +2nPi where n is an integer..

Btw, that was an example from my textbook...the first solution is what the textbook stated, and the second solution is what i thought right ..

Mspike6 said:
Woohoo got it ! ..

Sin2x + cos2x will be 1

so ..

it will be [2 + 2cosx] / [sinx + sinxcosx]

Cosx cancels each other

==> 4/ 2sinx

==> 2/sinx

==> 2cscx

Thanks a lot for the help guys :D

Mspike, don't make that mistake. The cosx don't divide out. Since you have already pretty much attained your answer, this is what should have happened:

$$\frac{2+2cosx}{sinx+sinxcosx}$$

$$=\frac{2(1+cosx)}{sinx(1+cosx)}$$ The (1+cosx) in the numerator and denominator divide out.

$$=\frac{2}{sinx}$$

$$=2cscx$$

Anakin_k said:
Mspike, don't make that mistake. The cosx don't divide out. Since you have already pretty much attained your answer, this is what should have happened:

$$\frac{2+2cosx}{sinx+sinxcosx}$$

$$=\frac{2(1+cosx)}{sinx(1+cosx)}$$ The (1+cosx) in the numerator and denominator divide out.

$$=\frac{2}{sinx}$$

$$=2cscx$$

Ahh!.
thanks for the heads up, and i thought i knew it :S

Mspike6 said:
I have anther question...

Find the general solutions for the following equations.

2 cos2 x -1 =0

Soultion.

Cos2x = 1/2
Cosx= - or + Sqrt 2 / 2

The general solution is pi /4 +2nPi, 3pi/4 +2nPi , 5Pi/4 + 2nPi and 7Pi/4 +2nPi , where n is an integer

But why can't i just take the Double angle of 2cos2x ?

which will be, cos2x=0

The general solution is Pi/2 +2nPi and 3Pi/2 +2nPi where n is an integer..
Btw, that was an example from my textbook...the first solution is what the textbook stated, and the second solution is what i thought right ..

In your solution, you did not even mention pi/4. The double angle is as follows:
2cos$$^{2}$$x-1=0
cos2x=0
2x=pi/2
x=pi/4

That is only one solution, and you are still missing it. Check where you went wrong.

Am sorry for asking so many questions ,

how one should factor , tan2-sec-1=0

i think it will be something like, (x-1)(x2+1)

where x and x2 are trig

Mspike6 said:
Am sorry for asking so many questions ,

how one should factor , tan2-sec-1=0

i think it will be something like, (x-1)(x2+1)

where x and x2 are trig

try using an identity involving tan2x and sec2x to replace the tan2x term.

Solve Tan2x-secx-1=0

Soultion

(Sec2x-1)-secx-1=0

sec2-secx-2=0

(secx-2)(secx+1)=0

--------------
Secx -2 =0
secx=2
x= Pi/3 , 5Pi/3

-------------
Secx+1=0
Secx=-1
x= Pi

X= Pi/3 , 5Pi/3, Pi

Is that right? am i missing something ?

Thank you :D

Mspike6 said:
Solve Tan2x-secx-1=0

Soultion

(Sec2x-1)-secx-1=0

sec2-secx-2=0

(secx-2)(secx+1)=0

--------------
Secx -2 =0
secx=2
x= Pi/3 , 5Pi/3

-------------
Secx+1=0
Secx=-1
x= Pi

X= Pi/3 , 5Pi/3, Pi

Is that right? am i missing something ?

Thank you :D

looks fine to me

Mspike6 said:
Solve Tan2x-secx-1=0

Soultion

(Sec2x-1)-secx-1=0

sec2-secx-2=0

(secx-2)(secx+1)=0

--------------
Secx -2 =0
secx=2
x= Pi/3 , 5Pi/3

-------------
Secx+1=0
Secx=-1
x= Pi

X= Pi/3 , 5Pi/3, Pi

Is that right? am i missing something ?

Thank you :D

Yup, if the domain was [0,2pi], then it is correct.

What are trigonometric identities in algebra?

Trigonometric identities in algebra are equations that involve trigonometric functions, such as sine, cosine, and tangent, and are true for all values of the variables involved. They are used to simplify and manipulate trigonometric expressions and equations.

What is the difference between a trigonometric identity and an equation?

A trigonometric identity is a mathematical statement that is true for all values of the variables involved, while an equation is a mathematical statement that is only true for specific values of the variables. Trigonometric identities are used to prove equations and to simplify expressions, while equations are used to solve for specific values.

How can I prove a trigonometric identity?

Trigonometric identities can be proven using algebraic manipulation, geometric reasoning, or the use of other known identities. It is important to start with one side of the identity and manipulate it until it is equal to the other side, using known identities and properties of trigonometric functions.

What are some common trigonometric identities?

Some common trigonometric identities include the Pythagorean identities (sin²θ + cos²θ = 1 and tan²θ + 1 = sec²θ), the double angle identities (sin2θ = 2sinθcosθ and cos2θ = cos²θ - sin²θ), and the sum and difference identities (sin(α ± β) = sinαcosβ ± cosαsinβ and cos(α ± β) = cosαcosβ ∓ sinαsinβ).

How can I use trigonometric identities to solve equations?

Trigonometric identities can be used to simplify equations involving trigonometric functions and to solve for specific values. By manipulating the given equation using known identities, it is possible to isolate the variable and find its value. This is particularly useful in solving trigonometric equations and verifying the solutions.

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