Find θ where 0 <= θ <= 2π for csc θ=(-2/√3)
Please answer in radian measure and without decimals, thank you
csc θ=(-2/√3)
or
sinθ= -√3/2 , where θ is in either III or IV
You should recognize the 30-60-90° triangle with corresponding sides of 1, √3, and 2
if sinθ = +√3/2, the θ = π/3 (60°)
so in quad III, θ = π + π/3 = 4π/3
and in IV, θ = 2π - π/3 = 5π/3
Thank you
Do you happen to know of any website or online tool hat I can use to test/graph this? I'm a visual learner and that would help a lot
Look at the "trig circle" in the top-right of the page.
https://www.google.ca/search?q=trig+circle+radians&rlz=1C6CHFA_enCA690CA691&sxsrf=ALeKk00eGYoXLNb1kxpyghssFekaiDJHjw:1590463648776&tbm=isch&source=iu&ictx=1&fir=7btvgKywl0C4yM%253A%252CPW_bt3de_fNtdM%252C_&vet=1&usg=AI4_-kR3sRiH6E_i9a_wW1ZzPOBKgIaIUw&sa=X&ved=2ahUKEwjmy8DmytDpAhWflHIEHeJhBC4Q_h0wAHoECAcQBA#imgrc=7btvgKywl0C4yM:
the ordered pairs are such that the x is the cosine of the angle and the y is the sine of the angle
e.g. for our problem , find our answers of θ = 4π/3 in quad III and in IV, θ = 5π/3
the y value of the ordered pairs will be -√3/2
If you click on the circle it will open a new page with some interesting stuff
To find the value of θ where csc θ = -2/√3, we first need to recall the definition of csc θ.
Cosecant (csc) is the reciprocal of sine. Therefore, csc θ = 1/sin θ.
We know that csc θ = -2/√3, so we can rewrite it as 1/sin θ = -2/√3.
Now, let's solve for sin θ.
Multiply both sides of the equation by sin θ to isolate it:
1 = -2sin θ/√3.
To get rid of the fraction, multiply both sides of the equation by √3:
√3 = -2sin θ.
Divide both sides by -2 to solve for sin θ:
sin θ = -√3/2.
To determine the value of θ, we need to find which angle(s) have a sine equal to -√3/2. To do this, we refer to the values of sine on the unit circle.
On the unit circle, the sine of an angle θ is equal to the y-coordinate of the corresponding point on the unit circle.
The values of sine for common angles in the first revolution (0 ≤ θ ≤ 2π) are as follows:
sin(0) = 0
sin(π/6) = 1/2
sin(π/4) = √2/2
sin(π/3) = √3/2
sin(π/2) = 1
sin(2π/3) = √3/2
sin(3π/4) = √2/2
sin(5π/6) = 1/2
sin(π) = 0
sin(7π/6) = -1/2
sin(5π/4) = -√2/2
sin(4π/3) = -√3/2
sin(3π/2) = -1
sin(5π/3) = -√3/2
sin(7π/4) = -√2/2
sin(11π/6) = -1/2
sin(2π) = 0
From the values above, we see that sin(π/3) = √3/2, and we are looking for sin θ = -√3/2.
By comparing these values, we can determine that θ equals either π/3 or 2π/3 in the given range (0 ≤ θ ≤ 2π).
Thus, the values of θ satisfying csc θ = -2/√3 are θ = π/3 and θ = 2π/3.