How do I solve a trig equation with two functions on one side?

  • Thread starter Hypochondriac
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In summary, the cosine and sine of x=-60 can be solved for x and x+180, but the trig functions need to be in cos or sin.
  • #1
Hypochondriac
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I have to solve:

2sinx = cos(x-60), for 0<=x<=360

so far i expanded the cos part

2sinx = cosxcos60 + sinxsin60

as cos60 = 1/2 and sin60 = (sqrt3)/2

2sinx = (1/2)cosx + ((sqrt3)/2)sinx

sinx = cosx + sqrt3 sinx

(1-sqrt3)sinx = cosx

but here's where I'm stuck, i have 2 trig functions in one equation and therefore cannot solve.
 
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  • #2
Hypochondriac said:
I have to solve:

2sinx = cos(x-60), for 0<=x<=360

so far i expanded the cos part

2sinx = cosxcos60 + sinxsin60

as cos60 = 1/2 and sin60 = (sqrt3)/2

2sinx = (1/2)cosx + ((sqrt3)/2)sinx

sinx = cosx + sqrt3 sinx
This line is incorrect. You have multiplied both sides by 2 to remove the factor of 1/2 on the right hand side, so the left hand side should be premultipled by 4.
 
  • #3
ahh yes, how silly of me,

but even now i have:

(4 - sqrt3)sinx = cosx

i need it all in sins or cos'
 
  • #4
Hypochondriac said:
i need it all in sins or cos'

Or, you could remember that tanx=sinx/cosx
 
  • #5
but if i divide through by cosx, i'll loose that solution of x because the cos' will cancel, its an equation not an expression.
 
  • #6
No you won't. There's only going to be one solution to that equation in the range specified.
 
  • #7
the answer in the back of the textbook gave 2 solutions.
I was told to never divide through with an equation, only with an expression

perhaps you only got one solution because you divided through to solve.
 
Last edited:
  • #8
Hypochondriac said:
perhaps you only got one solution because you divided through to solve.

Sorry, I am being really stupid! You don't lose a solution by dividing by cos(x), but of course the function tan(x)= 1/(4-sqrt(3)), is periodic with period 180 degrees. So, the solutions to this in the given range will be the principal value for arctan(1/(4-sqrt(3))) [the one given by your calculator], and this value with 180 added on.
 
  • #9
ok so arctan 1/(4-sqrt3) gives me my principle 23.8, and then +180 to give 203.8, my secondary.

Solved!
cheers, I am a bit weary about dividing through but I am not going to argue with the outcome!
 
  • #10
It's fine because you know that an x where cos(x) is zero can't possibly be a solution (otherwise you have 4 - sqrt(3) = 0).
 
  • #11
If you really wanted to, when you had sin on one side and cos on the other side, you could square both sides (and possibly introduce extraneous roots), and use an identity for (sinx)^2 or (cosx)^2, changing it to a quadratic type. The method above is much easier.
 

What is a trig equation?

A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, tangent, etc. The goal is to solve for the value of the variable that makes the equation true.

What are the basic steps for solving a trig equation?

The basic steps for solving a trig equation are:
1. Isolate the trig function on one side of the equation.
2. Use inverse trig functions to solve for the variable.
3. Check the solutions by plugging them back into the original equation.
4. If the equation has multiple solutions, use the period of the trig function to find all possible solutions.

What are the common trig identities used to solve trig equations?

Some common trig identities used to solve trig equations are:
1. Pythagorean identities: sin²θ + cos²θ = 1 and tan²θ + 1 = sec²θ
2. Double angle identities: sin2θ = 2sinθcosθ and cos2θ = cos²θ - sin²θ
3. Sum and difference identities: sin(α ± β) = sinαcosβ ± cosαsinβ and cos(α ± β) = cosαcosβ ∓ sinαsinβ

What are some common mistakes to avoid when solving a trig equation?

Some common mistakes to avoid when solving a trig equation are:
1. Forgetting to check for extraneous solutions when using inverse trig functions.
2. Forgetting to use parentheses when substituting values into the equation.
3. Not simplifying fractions or using common denominators when adding or subtracting trig functions.
4. Forgetting to use the correct unit of measurement (degrees or radians) when working with inverse trig functions.

Can a trig equation have more than one solution?

Yes, a trig equation can have multiple solutions. This is due to the periodic nature of trigonometric functions. To find all possible solutions, use the period of the trig function to generate additional solutions. For example, if the equation has a solution of θ = 30°, then additional solutions can be found by adding or subtracting multiples of the period, such as θ = 30° ± 360°n, where n is an integer.

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