Find two solutions of the following equations, giving your answers in radians
sin ϴ = 1/2
cosec ϴ = 2sqrt(3)/3
cot ϴ = -1
In your trig repertoire , you should know the 3 basic trig values of angles 0°, 30°, 45°, 60°, 90°
by knowing the ratio of sides of the 30-60-90 and the 45-45-90 degree triangles.
This is very very important in any further study of trig.
I will do the only "hard" one:
csc Ø = 2√3/3
sinØ = 3/(2√3) , since cscØ = 1/sinØ
3/(2√3)
= 3/(2√3) * √3/√3
= 3√3/6
= √3/2 <---- recognize that sin 60° = √3/2
so sinØ = √3/2 ----> Ø = 60°
But the sine is positive in quads I and II
so Ø is also equal to 180°-60° = 120°
of course in radians
Ø = π/3 or Ø = 2π/3
In your trig repertoire , you should know the 3 basic trig values of angles 0°, 30°, 45°, 60°, 90°
by knowing the ratio of sides of the 30-60-90 and the 45-45-90 degree triangles. ..Make some flash cards, and memorize these NOW. Or, you can be a dead duck at the end of the course.
To find the solutions of the given equations, we first need to understand the trigonometric functions involved.
1. For the equation sin ϴ = 1/2, we know that sin ϴ represents the ratio of the length of the opposite side to the length of the hypotenuse in a right triangle. The value 1/2 corresponds to the angle π/6 or 30 degrees. Since the question asks for the answer in radians, we convert 30 degrees to radians: π/6.
So, one solution for sin ϴ = 1/2 is ϴ = π/6.
To find the second solution, we note that sin function has a periodicity of 2π. Therefore, we can find the second solution by adding 2π to the first solution:
ϴ = π/6 + 2π = 13π/6.
Hence, the two solutions are ϴ = π/6 and ϴ = 13π/6.
2. For the equation cosec ϴ = 2√3/3, we can rewrite it as sin ϴ = 1/(2√3/3) = √3/2. This is a well-known value for sinϴ, which occurs at an angle of π/3 or 60 degrees.
Converting 60 degrees to radians, we get π/3.
So, one solution for cosec ϴ = 2√3/3 is ϴ = π/3.
To find the second solution, we add 2π to the first solution:
ϴ = π/3 + 2π = 7π/3.
Hence, the two solutions are ϴ = π/3 and ϴ = 7π/3.
3. For the equation cot ϴ = -1, we know that cot ϴ is the reciprocal of the tangent function. To find the angle that satisfies this equation, we need to find the angle whose tangent is -1.
The angle with a tangent of -1 is π/4 or 45 degrees. Converting π/4 to radians, we get π/4.
So, one solution for cot ϴ = -1 is ϴ = π/4.
To find the second solution, we add π to the first solution:
ϴ = π/4 + π = 5π/4.
Hence, the two solutions are ϴ = π/4 and ϴ = 5π/4.
Therefore, the two solutions for each of the given equations, in radians, are:
For sin ϴ = 1/2: ϴ = π/6, 13π/6.
For cosec ϴ = 2√3/3: ϴ = π/3, 7π/3.
For cot ϴ = -1: ϴ = π/4, 5π/4.