Question regarding a trig equation

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In summary, Apostol uses the angle addition identity for cosine to derive the equation cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ) in the middle of the attached image, but does not explicitly state this in the previous information. It appears that he used the equation cos(x+y) = cosxcosy-sinxsiny to arrive at this result.
  • #1
rxh140630
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Homework Statement
cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ)
Relevant Equations
no equation given for this trig equation
See the attached image. Apostol gives cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ), in the middle of the picture, but previous info given does not state how he got this equation.

To me it looks like he used the equation cos(x+y) = cosxcosy-sinxsiny
 
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  • #2
If a moderator could please remove the thread I guess it actually was just using a previous equation that was given. I was too tired and understand why this is the result now.
 
  • #3
rxh140630 said:
Homework Statement:: cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ)
Relevant Equations:: no equation given for this trig equation

See the attached image. Apostol gives cos(nθ) = cos((n-1)θ)cos(θ) - sin((n-1)θ)sin(θ), in the middle of the picture, but previous info given does not state how he got this equation.

To me it looks like he used the equation cos(x+y) = cosxcosy-sinxsiny
A moderator may come along and remove this thread, as you wish. However, this formula comes directly from applying angle addition identities for cos. I guess you suspected as much.

##\cos(n\theta) = \cos((n-1)\theta+\theta)##

## = \cos((n-1)\theta)\cos(\theta)-\sin((n-1)\theta)\sin(\theta)##
 

1. What is a trig equation?

A trigonometric equation is an equation that involves trigonometric functions such as sine, cosine, tangent, etc. These equations are used to model various real-life phenomena, such as waves and oscillations.

2. How do you solve a trig equation?

To solve a trig equation, you need to use the properties and identities of trigonometric functions, as well as algebraic techniques. You may also need to use a calculator or reference tables to find the values of trigonometric functions at specific angles.

3. What is the domain of a trig equation?

The domain of a trig equation depends on the specific trigonometric function involved. For example, the domain of sine and cosine functions is all real numbers, while the domain of tangent and cotangent functions is all real numbers except for the values where the function is undefined (i.e. where the denominator is equal to 0).

4. Can a trig equation have multiple solutions?

Yes, a trig equation can have multiple solutions. This is because trigonometric functions are periodic, meaning they repeat their values after a certain interval. Therefore, there can be multiple angles that satisfy a given trig equation.

5. How are trig equations used in real life?

Trig equations are used in many fields, including physics, engineering, and astronomy. They are used to model and predict various phenomena, such as the motion of planets, the behavior of waves, and the design of structures. They are also used in navigation and surveying to calculate distances and angles.

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