Help with Trig Function: Sec(2x)csc(x)sin(2x) and C=cosx

In summary: Well done! In summary, to write sec(2x)csc(x)sin(2x) as a function of C, you first need to replace sin(x) with cos(x) using the identity sin(x)=√(1-cos^2(x)). Then, you can simplify the formula to 2C/(2C^2-1).
  • #1
Mark53
93
0

Homework Statement


Let C=cosx. Write sec(2x)csc(x)sin(2x) as a function of C.

The Attempt at a Solution



Am I on the right track

1/cos(2x) * 1/sin(x) * 2sin(x)cos(x)

1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

What would i do from here?
 
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  • #2
There is one ##\sin x## left in your formula that you need to convert to only using ##\cos x##.
Having done that, you only need to replace every ##\cos x## by ##C##, then simplify as much as possible.
 
  • #3
Mark53 said:

Homework Statement


Let C=cosx. Write sec(2x)csc(x)sin(2x) as a function of C.

The Attempt at a Solution



Am I on the right track

1/cos(2x) * 1/sin(x) * 2sin(x)cos(x)

1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

What would i do from here?

So you got ##\frac{1}{\cos^2(x)-\sin^2(x)}\frac{1}{\sin(x)}2\sin(x)\cos(x)=\frac{2sin(x)\cos(x)}{(\cos^2(x)-\sin^2(x))\sin(x)}##

Why don't you simplify with sin(x)?
 
  • #4
andrewkirk said:
There is one ##\sin x## left in your formula that you need to convert to only using ##\cos x##.
Having done that, you only need to replace every ##\cos x## by ##C##, then simplify as much as possible.

1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

1/(cos^2(x)-(1-cos^2(x) * 1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

simplifying this I get

1/(cos(x)-1)

=1/(C-1)

Is this the correct answer?
 
  • #5
Mark53 said:
1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

1/(cos^2(x)-(1-cos^2(x) * 1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x)

simplifying this I get

1/(cos(x)-1)

=1/(C-1)

Is this the correct answer?

made a mistake it should be

2C/(2C^2-1)
 
  • #6
Mark53 said:
made a mistake it should be

2C/(2C^2-1)
Finally, that is correct.
 
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Likes Mark53

Related to Help with Trig Function: Sec(2x)csc(x)sin(2x) and C=cosx

1. What is the purpose of the trigonometric function "sec(2x)csc(x)sin(2x)"?

The purpose of this trigonometric function is to simplify and manipulate trigonometric expressions involving the sine, cosine, and secant functions. It can also be used to solve trigonometric equations and evaluate trigonometric identities.

2. How do you find the value of "sec(2x)csc(x)sin(2x)"?

To find the value of this function, you can use the trigonometric identities and properties to simplify the expression. Then, you can use a calculator or trigonometric tables to find the numerical value.

3. What is the relationship between "sec(2x)csc(x)sin(2x)" and the cosine function?

The relationship between these two functions is that sec(2x)csc(x)sin(2x) = csc(x)sin(2x)/cos(2x). This shows that the expression is related to the cosine function through the reciprocal identity csc(x) = 1/sin(x).

4. How can "sec(2x)csc(x)sin(2x)" be used to find the value of "C=cosx"?

To find the value of C=cosx using this function, you can first substitute C for cosx in the expression. Then, use trigonometric identities to simplify the expression and solve for C.

5. What are some common applications of "sec(2x)csc(x)sin(2x)" in science and mathematics?

This function is often used in physics, engineering, and mathematics to model and solve problems involving periodic motion, such as waves, vibrations, and oscillations. It is also used in calculus to evaluate integrals and derivatives involving trigonometric functions.

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