Solve the equation for the indicated domain.
sec(theta+81 degrees)=2
Theta E[0,360)
sec(Ø + 81) = 2
cos(Ø + 81) = 1/2
I know cos 60° = 1/2
so Ø+81 = 60 or Ø+81 = 300°
Ø = 60-81 = -21° or Ø = 219°
now the period of cos(Ø+81) = 360°
so adding 360 will give me another answer
so Ø = -21 + 360° = 339°
so in the given domain:
Ø = 219° , 339°
To solve the equation sec(theta+81 degrees) = 2 for the indicated domain [0,360), we need to isolate theta.
Step 1: Rewrite the equation
sec(theta+81 degrees) = 2
Step 2: Take the reciprocal of both sides
1 / sec(theta+81 degrees) = 1 / 2
Step 3: Rewrite the left side using the definition of secant
1 / sec(theta+81 degrees) = cos(theta+81 degrees)
Step 4: Invert the fraction on the right side
cos(theta+81 degrees) = 1/2
Step 5: Take the inverse cosine (cos^-1) of both sides to solve for theta
theta+81 degrees = cos^-1(1/2)
Step 6: Use the unit circle or a calculator to find the angle whose cosine is 1/2. The possible values are 60 degrees and 300 degrees, which correspond to theta+81 degrees.
Step 7: Solve for theta
theta+81 degrees = 60 degrees or theta+81 degrees = 300 degrees
Step 8: Subtract 81 degrees from both sides in each equation
theta = 60 degrees - 81 degrees or theta = 300 degrees - 81 degrees
Step 9: Simplify
theta = -21 degrees or theta = 219 degrees
Since the indicated domain is [0,360), the solutions within this range are theta = 219 degrees.
To solve the equation sec(theta+81 degrees) = 2 in the indicated domain of theta within the interval [0, 360), you can follow these steps:
Step 1: Simplify the equation using trigonometric identities.
The reciprocal identity for secant is: sec(x) = 1/cos(x). Applying this identity to the equation, we have: 1/cos(theta + 81 degrees) = 2.
Step 2: Isolate the cosine term.
To do this, multiply both sides of the equation by cos(theta + 81 degrees): 1 = 2*cos(theta + 81 degrees).
Step 3: Solve for the angle term.
Divide both sides of the equation by 2: cos(theta + 81 degrees) = 1/2.
Step 4: Use inverse cosine function.
We have cos(theta + 81 degrees) = 1/2. To find the angle theta, we need to take the inverse cosine (arccos) of both sides of the equation: theta + 81 degrees = arccos(1/2).
Step 5: Find the principal angle.
The arccosine of 1/2 is 60 degrees or π/3 radians. So we have: theta + 81 degrees = 60 degrees.
Step 6: Solve for theta.
Subtract 81 degrees from both sides of the equation: theta = 60 degrees - 81 degrees = -21 degrees.
Step 7: Adjust theta within the given domain.
Since the domain for theta is [0, 360), the angle -21 degrees is not within this range. To find a solution within the given domain, we need to add the period of the cosine function (360 degrees) to -21 degrees.
theta = -21 degrees + 360 degrees = 339 degrees.
Therefore, the solution to the equation sec(theta+81 degrees) = 2, with theta E [0,360), is theta = 339 degrees.