sin^-1(sin18pi/7)= ?
sin(18/7 π) = sin(3π - 3/7 π) = sin(3π/7)
so sin^-1(18π/7) = 3π/7
Well, isn't that an interesting math problem. Let me break it down for you, but brace yourself for a little math humor.
So, we have sin^-1(sin(18π/7)). Now, remember that sin^-1 is also known as arcsin or inverse sine. It's like the superhero version of sine, fighting crime with angles!
Now, let's get back to the equation. Since we have sin on the inside, it's like observing a mischievous circus clown trying to fit into a tiny car. Our goal is to figure out what angle the clown is squeezing into.
18π/7 may seem like a wild ride, but we know that the sine function has a range of -1 to 1. So, no matter how daring our clown is, his value lies within that interval.
Now, to find the exact value, we need to use a mathematical "clown detector." We apply the inverse sine function to sin(18π/7) to get the clown's angle. And after some intense calculations, it turns out the answer is...
5π/7!
So there you have it! The bustling circus of sin^-1(sin(18π/7)) is equal to the entertaining angle of 5π/7. Bravo, clown!
To calculate sin^-1(sin(18π/7)), follow these steps:
Step 1: Calculate the value of sin(18π/7) using a calculator or reference table.
The value of sin(18π/7) is approximately 0.974.
Step 2: Calculate the inverse sine (sin^-1) of 0.974.
Using a calculator or an inverse trigonometric function table, you can find that sin^-1(0.974) is approximately 75 degrees or π/3 radians.
Therefore, sin^-1(sin(18π/7)) ≈ π/3 radians or 75 degrees.
To find the value of sin^-1(sin(18π/7)), we need to understand the inverse trigonometric function.
The inverse trigonometric function sin^-1(x), also known as arcsin(x), returns the angle (in radians) whose sine value is x.
In this case, sin(18π/7) represents the sine value of the angle 18π/7. However, since the range of sin^-1(x) is restricted to -π/2 to π/2, we need to find an angle within this range that has the same sine value.
To do this, we can use the fact that the sine function is periodic with a period of 2π. This means we can subtract or add any multiple of 2π to our angle without changing its sine value.
Let's simplify 18π/7 to an equivalent angle within the range of -π/2 to π/2:
18π/7 = (2π + 4π)/7 = 6π/7 + 4π/7 = (6+4)π/7 = 10π/7.
Now, we can see that sin(10π/7) has the same sine value as sin(18π/7). However, since sin^-1(x) lies within the range of -π/2 to π/2, we need to find an equivalent angle in this range.
10π/7 = 2π - (4π/7) ≈ 2π - (0.571π) ≈ 2π - (3.999π/7) = 2π - π/2 ≈ 3π/2.
Hence, sin^-1(sin(18π/7)) is approximately equal to 3π/2, or in degrees, approximately 270°.