Solving trig equations using Addition formula

Use the double-angle formula for tangent which is tan(2a) = (2 tan(a))/(1-tan^2(a)) Lastly use the inverse tangent function.In summary, to solve the equation sin(x+∏/6)=2cosx for 0≤x≤2∏, you can subtract (1/2)cosx from both sides and then use the double-angle formula for tangent to find the value of x.
  • #1
studentxlol
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0

Homework Statement



Solve the equation: sin(x+∏/6)=2cosx for 0≤x≤2∏

Homework Equations



sinAcosB+cosAsinB

The Attempt at a Solution



sin(x+∏/6)=2cosx

2cosx=(√3)/2sinx+1/2cosx

How do I solve from here?
 
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  • #2
Moved the thread to precalc. Please post in the correct forum next time.
 
  • #3
Hi studentxlol! :smile:

Perhaps you can use the following trig identity?

a sin x + b cos x = √(a2+b2) sin(x+ϕ)

where cos ϕ = a/√(a2+b2)
and sin ϕ= b/√(a2+b2)
 
  • #4
studentxlol said:

Homework Statement



Solve the equation: sin(x+∏/6)=2cosx for 0≤x≤2∏

Homework Equations



sinAcosB+cosAsinB

The Attempt at a Solution



sin(x+∏/6)=2cosx

2cosx=(√3)/2sinx+1/2cosx

How do I solve from here?

The first, obvious, step is to subtract (1/2) cos(x) from both sides. Then get a tangent.
 

FAQ: Solving trig equations using Addition formula

What is the Addition formula in trigonometry?

The Addition formula in trigonometry is a mathematical rule that allows us to add or subtract two trigonometric functions to create a new function.

How do you solve trig equations using the Addition formula?

To solve trig equations using the Addition formula, you need to follow these steps:

  • Identify the trig functions involved in the equation.
  • Use the Addition formula to rewrite the equation in terms of a single trig function.
  • Solve the new equation using algebraic techniques.
  • Check your solution by substituting it back into the original equation.

Can the Addition formula be used for all trigonometric functions?

Yes, the Addition formula can be used for all trigonometric functions, including sine, cosine, tangent, cotangent, secant, and cosecant.

Are there any special cases when using the Addition formula?

Yes, there are two special cases when using the Addition formula:

  • If the trig functions in the equation have different arguments (angles), then you need to use the Subtraction formula instead.
  • If the trig functions have the same argument, you can simplify the equation using the Pythagorean identity before applying the Addition formula.

How do you know when to use the Addition formula?

You can use the Addition formula when you have an equation that involves adding or subtracting two trigonometric functions, such as sin(x + y) or cos(x - y). If the equation involves multiplication or division of trig functions, you may need to use other trigonometric identities instead.

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