find all values of theta in the interval 0<theta<360 that satisfy the equation 3cos 2theta + 2 sin theta+1=0
3cos^2 Ø + 2sinØ + 1 = 0
3(1 - sin^2 Ø) + 2sinØ + 1 = 0
3sin^2Ø - 2sinØ -4 = 0
sinØ = (2 ± √28)/6
= 1.215.. which is not admissable
or
sinØ = -.54858...
so Ø is in III or IV
the angle in standard position is 33.27°
so Ø = 213.27° or Ø = 326.73°
To solve the equation 3cos(2θ) + 2sin(θ) + 1 = 0, we can use some trigonometric identities to simplify it.
First, we'll rewrite sin(θ) in terms of cos(θ) using the identity sin²(θ) + cos²(θ) = 1.
2sin(θ) = 2 * √(1 - cos²(θ))
Now, let's substitute this into the equation:
3cos(2θ) + 2sin(θ) + 1 = 0
3cos(2θ) + 2 * √(1 - cos²(θ)) + 1 = 0
To simplify further, we'll use the double angle formula for cosine: cos(2θ) = 2cos²(θ) - 1.
Substituting cos(2θ) = 2cos²(θ) - 1 into the equation:
3(2cos²(θ) - 1) + 2 * √(1 - cos²(θ)) + 1 = 0
6cos²(θ) - 3 + 2 * √(1 - cos²(θ)) + 1 = 0
6cos²(θ) + 2 * √(1 - cos²(θ)) - 2 = 0
Now, let's solve for cos(θ):
6cos²(θ) + 2 * √(1 - cos²(θ)) - 2 = 0
Divide the entire equation by 2:
3cos²(θ) + √(1 - cos²(θ)) - 1 = 0
Let's make a substitution: Let u = cos(θ):
3u² + √(1 - u²) - 1 = 0
This gives us a quadratic equation in terms of u. By solving this quadratic equation, we can find values of u (cos(θ)), and then solve for θ.
Unfortunately, solving this equation requires complex mathematical calculations and cannot be done step-by-step. However, you can use graphing calculators or numerical methods to find the solutions.
To find all values of θ in the interval 0° < θ < 360° that satisfy the equation 3cos(2θ) + 2sin(θ) + 1 = 0, we can use trigonometric identities and equations to simplify and solve the equation step by step.
Step 1: Use the double-angle identity for cosine.
The double-angle identity for cosine states that cos(2θ) = cos²(θ) - sin²(θ).
Substituting this into the equation:
3[cos²(θ) - sin²(θ)] + 2sin(θ) + 1 = 0
Step 2: Simplify the equation.
Rearranging terms and combining like terms:
3cos²(θ) - 3sin²(θ) + 2sin(θ) + 1 = 0
Step 3: Apply the Pythagorean identity.
The Pythagorean identity states that cos²(θ) + sin²(θ) = 1.
Substituting this into the equation:
3(1 - sin²(θ)) - 3sin²(θ) + 2sin(θ) + 1 = 0
Step 4: Simplify and rearrange the equation.
Expanding and rearranging terms:
3 - 3sin²(θ) - 3sin²(θ) + 2sin(θ) + 1 = 0
-6sin²(θ) + 2sin(θ) + 4 = 0
Step 5: Factor the equation.
Factoring the equation:
-2(3sin²(θ) - sin(θ) - 2) = 0
Step 6: Solve for sin(θ).
Now we can solve the quadratic equation: 3sin²(θ) - sin(θ) - 2 = 0.
Using factoring or the quadratic formula, we can find that:
(3sin(θ) + 2)(sin(θ) - 1) = 0
Setting each factor equal to zero and solving for θ:
3sin(θ) + 2 = 0
sin(θ) = -2/3
sin(θ) - 1 = 0
sin(θ) = 1
Step 7: Find the values of θ that satisfy sin(θ) = -2/3.
Using the unit circle or a calculator, we find the angles where sin(θ) = -2/3 are approximately -41.8° and 221.8°.
Step 8: Find the values of θ that satisfy sin(θ) = 1.
Using the unit circle or a calculator, we find the angle where sin(θ) = 1 is 90°.
In conclusion, the values of θ in the interval 0° < θ < 360° that satisfy the equation 3cos(2θ) + 2sin(θ) + 1 = 0 are -41.8°, 90°, and 221.8°.