How do you solve: arcsin(sin 5pi/4)
That is not Calculus.
sin ( 5 π / 4 ) = sin ( 4 π / 4 + π / 4 ) = sin ( π + π / 4 )
Since:
sin ( π + θ ) = - sin θ
sin ( 5 π / 4 ) = sin ( π + π / 4 ) = - sin ( π / 4 ) = - 1 / √ 2
The range of sin x :
−1 ≤ sin x ≤ 1
sin ( - π / 2 ) = - 1 , sin ( π / 2 ) = 1
So x must be in interval:
x ∈ [ - π / 2, π / 2 ]
The only angle in this interval whose sine is - 1 / √ 2 is x = - π / 4
because - π / 2 < - π / 4 < π / 2
So the solution is:
x = - π / 4
arcsin [ sin ( 5 π / 4 ) ] = - π / 4
To solve the expression "arcsin(sin(5π/4))," follow these steps:
Step 1: Simplify the inside function.
- The sine function is applied to the angle 5π/4.
- Since the sine function is periodic with a period of 2π, we subtract 2π from the given angle until it falls within the range [-π/2, π/2].
- To bring 5π/4 within the range [-π/2, π/2]:
- First, subtract 2π: 5π/4 - 2π = 5π/4 - 8π/4 = -3π/4
- This new angle, -3π/4, falls within the range [-π/2, π/2].
Step 2: Compute the arcsine of -3π/4.
- The arcsine (or inverse sine) of -3π/4 can be found using a calculator or a trigonometric identity.
- Using a calculator: arcsin(-3π/4) ≈ -0.84806207898
- If you are required to provide the answer in radians, this is the final result. If the answer should be in degrees, continue to the next step.
Step 3: Convert the answer to degrees (if necessary).
- To convert radians to degrees, multiply the radian measure by 180 and divide by π.
- Converting -0.84806207898 radians to degrees:
- Multiply: -0.84806207898 * (180/π) ≈ -48.571253638 degrees rounded to the nearest degree.
- Therefore, arcsin(sin(5π/4)) ≈ -48° (rounded to the nearest degree).
To solve the equation arcsin(sin(5π/4)), there are a few steps you can follow:
1. Identify the domain of the arcsin function: The arcsin function takes a value between -1 and 1 and returns an angle between -π/2 and π/2.
2. Simplify the inside function: In this case, the inside function is sin(5π/4). We can determine the value of sin(5π/4) by using the unit circle or by evaluating the sin function at 5π/4.
- Using the unit circle, we find that 5π/4 represents a point in the third quadrant, where the sin function is negative. Thus, sin(5π/4) = -√2/2.
- Alternatively, evaluating the sin function at 5π/4, sin(5π/4) = sin(π + π/4) = -sin(π/4) = -√2/2.
3. Plug the value -√2/2 into the arcsin function: arcsin(-√2/2).
4. Solve for the angle: To find the angle, you can use a calculator or reference tables that provide the arcsin values for certain angles.
- In this case, arcsin(-√2/2) ≈ -π/4.
Therefore, the solution to the equation arcsin(sin(5π/4)) is approximately -π/4.