find all solutions of each equations
a) cos2x=square root of 3/2
b)cos 2theta/3 =-1
c) cot 3 theta /2 = -square root of 3
d) sin (2x-pie/4) = square root of 2/2
You should have a common method of attack for these kind of problems
What do you find difficult about them?
I will do the first, you try the same approach for the others.
cos 2Ø = √3/2
since the cosine is positive, we know that 2Ø must be in either quadrant I or quad V, by the CAST rule.
using my calculator, (or knowing the ratio of the 3-060-90 triangle),
we know cos 30 = √3/2
so 2Ø = 30° in I or 2Ø = 360-30 = 330° in IV
then Ø = 15° or Ø = 165° as our initial solutions
which in radians would be π/12 or 11π/12
Since you wanted "all solutions" , we look at the period of cos 2Ø
since the period of cos 2Ø = 360/2 = 180° or π radians, we add/subtract integer multiples of 18° to each answer,
general solution:
Ø = 15° +180k° , or 165+ 180k , where k is an integer
in radians,
Ø = π/12 + kπ or 11π/12 + kπ
Sure! Let's find the solutions to each of these equations:
a) cos(2x) = √3/2
To find the solutions, we need to look for the values of x that satisfy this equation. One way to approach this is to find the angles for which cos(2x) equals √3/2. We know that cos(60°) = √3/2, so we can write:
2x = 60° + 360°n and 2x = 300° + 360°n, where n is an integer.
From these equations, we can solve for x:
x = 30° + 180°n and x = 150° + 180°n, where n is an integer.
These are the solutions to the given equation.
b) cos(2θ/3) = -1
Here, we need to find the values of θ that satisfy the equation. We know that cos(180°) = -1, so we can write:
2θ/3 = 180° + 360°n, where n is an integer.
Solving for θ:
θ = 270° + 540°n, where n is an integer.
These are the solutions to the given equation.
c) cot(3θ/2) = -√3
To find the solutions, we need to look for the values of θ that satisfy this equation. We know that cot(60°) = √3, so we can write:
3θ/2 = 60° + 180°n, where n is an integer.
Solving for θ:
θ = 40° + 120°n, where n is an integer.
These are the solutions to the given equation.
d) sin(2x - π/4) = √2/2
Here, we need to find the values of x that satisfy the equation. We know that sin(45°) = √2/2, so we can write:
2x - π/4 = 45° + 360°n, where n is an integer.
Solving for x:
x = (45° + π/4)/2 + 180°n, where n is an integer.
Note: π is the symbol for pi.
These are the solutions to the given equation.
I hope this helps! Let me know if you have any further questions.