Question

Solve the equation on the interval 0 is less than or equal to theta.Cos (2theta) + 14 sin ²theta =10

Answer 1

To solve the equation cos(2theta) + 14sin^2(theta) = 10 on the interval 0 ≤ theta, follow these steps: rearrange and combine terms, use the double angle formula for cosine, simplify the equation, solve the quadratic equation, take the square root, and use inverse trigonometric functions to find the values of theta within the given interval.

Step-by-step explanation:

To solve the equation cos(2theta) + 14sin^2(theta) = 10 on the interval 0 ≤ theta, we can use algebraic manipulation and trigonometric identities. Here are the steps:

1. Start by rearranging the equation and combining like terms to get 14sin^2(theta) = 10 - cos(2theta).
2. Using the double angle formula for cosine, rewrite cos(2theta) as 1 - 2sin^2(theta).
3. Substitute this value back into the equation and simplify to get 14sin^2(theta) = 10 - 1 + 2sin^2(theta).
4. Combine like terms and rearrange to get a quadratic equation in terms of sin^2(theta). Solve this quadratic equation to find the possible values of sin^2(theta).
5. Take the square root of these values to find the possible values of sin(theta).
6. Finally, use the inverse trigonometric function to find the values of theta within the given interval.