Trig Identity Question Sort of

In summary, the objective of the conversation was to express y(t) = cos(t - b) - cos(t) in the form y(t) = Asin(t - c), where A and c are in terms of b. The conversation used the sum and difference formulas to get the desired expression and then solved for A and c by equating the two expressions. The final equations for A and c were A = +/- sqrt(-2cosb + 1) and c = arcsin(A/2). The conversation ended with the acknowledgement that this approach was correct and the next step was to divide the two equations to solve for tan(c).
  • #1
s_j_sawyer
21
0

Homework Statement



Okay so the objective here is to express

y(t) = cos(t - b) - cos(t)

in the form

y(t) = Asin(t - c)

where A and c are in terms of b.

Homework Equations



For easy reference, here is a table of identities:
http://www.sosmath.com/trig/Trig5/trig5/trig5.html

The Attempt at a Solution



Well, using the sum and difference formulas, I got that

y(t) = cost(cosb - 1) + sint*sinb

equating this to the desired expression gives

cost(cosb - 1) + sint*sinb = Asin(t - c)
cost(cosb - 1) + sint*sinb = A(sint)(cosc) - A(cost)(sinc)

So thus I determined that

cosb - 1 = -Asinc (1)
sinb = Acosc (2)

Squaring both sides and adding gave me, eventually,

A^2 = -2cosb + 1

So would A be +/- sqrt(-2cosb + 1) ?

Then I did almost the exact same thing for c simply by moving the -1 on the left side of (1) to the right:

cosb = -Asinc + 1 (1*)
sinb = Acosc (2)

Squaring and adding I got

A^2 - 2Asinc = 0

A - 2sinc = 0

sinc = A/2

so then would c = arcsin(A/2)?I don't even know if I am doing this right so any assistance would be great!

Thank you.
 
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  • #2
cost(cosb - 1) + sint*sinb = A(sint)(cosc) - A(cost)(sinc)

cosb - 1 = -Asinc (1)
sinb = Acosc (2)

Squaring and adding I got

A^2 - 2Asinc = 0

It is very hard to read it but squaring and ading (1) and (2) should give A^2 on the right side but looks like you simplified.

Yes, you are using the right approach. In case you missed second step is to divide (1) by (2) to get second equation.
A.tan(c) = ...
 

1. What are trigonometric identities?

Trigonometric identities are equations that show the relationship between different trigonometric functions, such as sine, cosine, and tangent. They are used to simplify and solve trigonometric equations.

2. Why are trigonometric identities important?

Trigonometric identities are important because they allow us to rewrite complex trigonometric expressions in simpler forms, making it easier to solve equations and perform calculations.

3. How do you prove trigonometric identities?

There are several methods for proving trigonometric identities, including direct proof, proof by induction, and proof by contradiction. These methods involve manipulating equations using algebraic rules and trigonometric identities to show that both sides of the equation are equal.

4. What is the difference between an identity and an equation?

An identity is an equation that is true for all values of the variables, while an equation is only true for specific values of the variables. In other words, an identity is always true, while an equation may or may not be true depending on the values of the variables.

5. How can trigonometric identities be used in real-world applications?

Trigonometric identities are used in a variety of fields, such as engineering, physics, and navigation. They can be used to solve real-world problems involving angles, distances, and heights, and are especially useful in geometric and trigonometric applications.

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