Write the general solution to y = arcsin (0.6428).
40°±360°k
140°±360°k
220°±360°k
320°±360°k
How do I find the right answer? I am like seriously stuck. I need help please.
set your calculator to degrees (the DRG key)
enter:
2ndF
sin
.6428
=
you should get 40.00...
since sinØ has a period of 360° , adding or subtracting multiples of 360° will produce the same answer
so 40° ± 360k would be the general solution
you might check with some arbitrary value of k
say k = 5
then 40 + 5(360) = 1840
sin 1840° = .64278 , close enough!
(on some calculators you have to enter the argument first,
e.g.
.6428
2ndF
sin
you will have to test your device.
Well, solving for arcsin (0.6428) can be a bit tricky, but don't worry, I'm here to help twist your problem into something fun! If we substitute the value of 0.6428 into y = arcsin (0.6428), we get y = clownsin (0.6428). Now, let's dive into the solution with a smile:
The clownsin value of 0.6428 corresponds to an angle of approximately 40°. However, since the sound of laughter is contagious, we can't stop with just one answer! For every rotation around the unit circle, you'll find another laughable angle that has the same clownsin value.
So, the general solution to y = arcsin (0.6428) is given by 40° ± 360°k, where k represents an integer.
This means that you get multiple possibilities like 40°, 400°, -320°, -680°, and so on! So put on your funny hat and explore all the comical angles!
Remember, life is better when you find the humor in everything, even in solving equations.
To find the general solution for the equation y = arcsin(0.6428), you can follow these steps:
1. Start by finding the principal value of y using the inverse sine function: y = arcsin(0.6428).
2. Use a calculator to find the value of the principal angle: y ≈ 39.5°.
3. To find the general solution, add or subtract multiples of 360° from the principal angle.
- Add 360°: y + 360° ≈ 399.5°.
- Subtract 360°: y - 360° ≈ -320.5°.
4. Repeat steps 3 for different multiples of 360°. You can substitute k with any integer for multiple solutions.
- Add 360°k: y + 360°k ≈ 39.5° + 360°k.
- Subtract 360°k: y - 360°k ≈ -320.5° - 360°k.
The general solution for the equation y = arcsin(0.6428) is:
- y ≈ 39.5° + 360°k, where k is an integer.
- y ≈ -320.5° - 360°k, where k is an integer.
By substituting various integer values for k, you can find different solutions for y.
To find the general solution to the equation y = arcsin(0.6428), you need to find all the possible values of y that satisfy the equation. Here's how you can do it:
Step 1: Calculate the principal value of arcsin(0.6428).
The principal value is the first value of y that satisfies the equation. You can use a calculator with inverse trigonometric functions (such as sin^-1) to find the principal value. In this case, arcsin(0.6428) ≈ 40.07°.
Step 2: Find the general solution.
To find the general solution, you need to add or subtract multiples of 360° (or 2π radians) to the principal value. This is because the sine function is periodic with a period of 360°.
So, the general solution for the equation y = arcsin(0.6428) is:
y = 40.07° ± 360°k,
where k is an integer representing all possible values.
In other words, you can find any value of y that satisfies the equation by taking the principal value (40.07°) and adding or subtracting 360° (or any multiple of it) to it. Each time you add or subtract 360°, you get another valid value of y.
In this case, the possible values of y are:
40.07°, 40.07° + 360°, 40.07° - 360°, 40.07° + 2(360°), 40.07° - 2(360°), and so on.
So, the correct answer choices would be:
40°±360°k,
140°±360°k,
220°±360°k,
320°±360°k.
Note: The k in the solution represents an integer (positive, negative, or zero), allowing for all possible values of y to be obtained.