Find all solutions between 0 and 2pi. Round to two decimal places. In radians.
Find all solutions between 0 and 2 pi. Round to two decimal places for the final solutions. The answers should be in radian mode. If you can use exact values use them. At least one of the 3 problems can use exact values for the answers and work to get to the answers.
2. 150sin3s=27
3. 16cos(3t-1)+4=-8
4. 9sin(pi/2(t-1))+5=-1
2.
sin 3s = 27/150
3s = .18198 in I or 3s = 2.96 in II
s = .06 or s = .99
but period of sin 3s = 2π/3
so by adding 2π/3 to any existing answer yields a new answer
x = .06 , 2.15 , 4.25
x = .99 , 3.08, 5.18
3.
16 cos(3t-1) = -12
cos(3t-1) = -12/16 = -.75
3t - 1 is in II or III
3t-1 = 2.42 or 3t-1 = 3.86
3t = 3.42 or 3t = 4.86
t = 1.14 or t = 1.62
again, period of cos(3t-1) = 2π/3
t = 1.14 , 3.23 , 5.33
t = 1.62 , 3.72 , 5.81
4.
9sin(pi/2(t-1)) = -6
sin(pi/2(t-1)) = -6/9 = -2/3
(π/2)(t-1) = 3.871 or (π/2)(t-1) = 5.55
t-1 = 2.46 or t = 3.54
t = 3.46 or t = 4.54
period = 2π/(π/2) = 4
but by adding 4 we go beyond the domain
so t = 3.46 or t = 4.54
Use the graph of the sine function to find all the solutions of the equation. (Enter your answer in the form a + bπn, where a [0, 2π), b is the smallest possible positive number, and n represents any integer.)
cos(t) = -1
I've tried everything to understand the problem. I have gotten to a point where I can get the correct answer but I do not understand how to put it in the form that it wants.
To solve each of these equations, we will follow the steps below:
2. 150sin3s = 27
Step 1: Divide both sides of the equation by 150:
sin3s = 27/150
sin3s = 0.18
Step 2: Take the inverse sine of both sides to isolate s:
3s = arcsin(0.18)
Step 3: Divide both sides by 3 to solve for s:
s = (1/3) * arcsin(0.18)
Step 4: Substitute the value of s into the equation:
s ≈ 0.19 radians
3. 16cos(3t-1) + 4 = -8
Step 1: Subtract 4 from both sides of the equation:
16cos(3t-1) = -12
Step 2: Divide both sides of the equation by 16:
cos(3t-1) = -12/16
cos(3t-1) = -0.75
Step 3: Take the inverse cosine of both sides to isolate (3t-1):
3t-1 = arccos(-0.75)
Step 4: Add 1 to both sides of the equation and divide by 3 to solve for t:
t = (arccos(-0.75) + 1)/3
Step 5: Substitute the value of t into the equation:
t ≈ 0.79 radians
4. 9sin(pi/2(t-1)) + 5 = -1
Step 1: Subtract 5 from both sides of the equation:
9sin(pi/2(t-1)) = -6
Step 2: Divide both sides of the equation by 9:
sin(pi/2(t-1)) = -6/9
sin(pi/2(t-1)) = -2/3
Step 3: Take the inverse sine of both sides to isolate (pi/2(t-1)):
pi/2(t-1) = arcsin(-2/3)
Step 4: Divide both sides of the equation by pi/2 and solve for (t-1):
t-1 = (2/pi) * arcsin(-2/3)
Step 5: Add 1 to both sides of the equation to solve for t:
t = 1 + (2/pi) * arcsin(-2/3)
Step 6: Substitute the value of t into the equation:
t ≈ 0.63 radians
Therefore, the solutions to the equations are:
2. s ≈ 0.19 radians
3. t ≈ 0.79 radians
4. t ≈ 0.63 radians
To find all solutions between 0 and 2π for each of the given equations, we will follow these steps:
1. Solve the equation algebraically to isolate the trigonometric function.
2. Use inverse trigonometric functions to solve for the angle.
3. Apply the periodicity of the trigonometric function to find all solutions within the given range.
4. Round the solutions to two decimal places, if necessary.
Let's solve each equation step by step:
2. 150sin(3s) = 27
Step 1: Divide both sides of the equation by 150 to isolate the sine function.
sin(3s) = 27/150
Step 2: Apply the inverse sine function (sin^(-1)) to both sides of the equation.
3s = sin^(-1)(27/150)
Step 3: Apply the periodicity of the sine function. Since we are looking for solutions between 0 and 2π, we add multiples of 2π to the angle until we obtain all solutions.
3s = sin^(-1)(27/150) + 2πn, where n is an integer
Step 4: Solve for s by dividing both sides by 3.
s = [sin^(-1)(27/150) + 2πn]/3
Now you can use a calculator to evaluate the exact or approximate values for sin^(-1)(27/150) (using the inverse sine function) and round the solutions to two decimal places within the desired range of 0 to 2π.
Follow the same steps for equations 3 and 4, replacing the trigonometric functions and constants as follows:
3. 16cos(3t - 1) + 4 = -8
Step 1: Subtract 4 from both sides of the equation.
16cos(3t - 1) = -12
Step 2: Divide both sides by 16.
cos(3t - 1) = -12/16
Step 3: Apply the inverse cosine function (cos^(-1)) to both sides.
3t - 1 = cos^(-1)(-12/16)
Apply the periodicity of the cosine function, add multiples of 2π to the angle until we obtain all solutions.
3t - 1 = cos^(-1)(-12/16) + 2πn, where n is an integer
Step 4: Solve for t by adding 1 and dividing by 3.
t = [cos^(-1)(-12/16) + 2πn + 1]/3
Again, calculate the exact or approximate value for cos^(-1)(-12/16) using a calculator, and round the solutions to two decimal places within the range 0 to 2π.
4. 9sin(π/2(t - 1)) + 5 = -1
Step 1: Subtract 5 from both sides of the equation.
9sin(π/2(t - 1)) = -6
Step 2: Divide both sides by 9.
sin(π/2(t - 1)) = -6/9
Simplify the fraction:
sin(π/2(t - 1)) = -2/3
Step 3: Apply the inverse sine function (sin^(-1)) to both sides.
π/2(t - 1) = sin^(-1)(-2/3)
Apply the periodicity of the sine function and solve for t:
t - 1 = [sin^(-1)(-2/3) + 2πn]/(π/2), where n is an integer
Step 4: Solve for t by adding 1.
t = [sin^(-1)(-2/3) + 2πn + 2]/(π/2)
Evaluate sin^(-1)(-2/3) using a calculator and round the solutions to two decimal places within the range 0 to 2π.
Remember to substitute the exact or approximated values for trigonometric functions using a calculator to compute the solutions accurately.