if -sinθ= 0.5736 and 90°≤θ≤180° , find the value of θ
-sinθ= 0.5736
sinθ= - 0.5736
90°≤θ≤180° <----- you are in quadrant II, in that quad the sine is positive,
so there is no solution to your equation
if you meant
sinθ= 0.5736
the angle in standard position would be 35.002° or appr 35°
so θ = 180° - 35° = 145°
Well, that's a tricky one. Let me get my clown calculator out to help me with the math... Hmm... Ah, here it is!
Now, since we know that -sinθ is equal to 0.5736, we can use the inverse sine function to find the value of θ.
So, according to my calculations, I've determined that θ is equal to approximately 53.1 degrees.
But please, take my answer with a grain of clown confetti. It's always good to double-check the math on your own!
To begin, we know that -sinθ = 0.5736.
To find the value of θ, we can use the inverse sine function (also known as arcsine or sin^-1). This will allow us to find the angle whose sine is equal to 0.5736.
Using a calculator, we can calculate the inverse sine of 0.5736, which gives us approximately 35.83°.
However, we need to remember that the given range for θ is 90° ≤ θ ≤ 180°.
Since 35.83° does not fall within this range, we need to find the angle that lies within the given range and has the same sine value.
As -sinθ has the same magnitude as sinθ, we need to find the angle within the given range where sinθ = 0.5736.
Using the calculator, we see that sinθ = 0.5736 when θ is approximately 35.83° or 180° - 35.83°, which is approximately 144.17°.
Since 90° ≤ θ ≤ 180°, the value of θ that satisfies -sinθ = 0.5736 is approximately 144.17°.
To find the value of θ, we can use the inverse sine function (also called arcsine).
Step 1: Write the given equation: -sinθ = 0.5736
Step 2: Apply the inverse sin (arcsine) function to both sides of the equation to isolate θ. The inverse sine function "undoes" the sin function, giving us the original angle.
arcsin(-sinθ) = arcsin(0.5736)
Step 3: Simplify the equation using the properties of the arcsine function. Since 90° ≤ θ ≤ 180°, we know that the sine function is negative in this range. Therefore, arcsin(-sinθ) will be equal to θ.
θ = arcsin(0.5736)
Step 4: Use a calculator to find the arcsin of 0.5736. Make sure your calculator is set to degrees mode.
θ ≈ 35.24°
Therefore, the value of θ is approximately 35.24° when -sinθ = 0.5736 and 90° ≤ θ ≤ 180°.