Solve each equation for exact solutions over the interval [0^∘, 360^∘) In Exercises 83-90, give

step 1 In this problem of technometric equation we have given that 4 this is 4 cost 2 theta is equal to

Solve each equation for exact solutions over the interval...

Solve each equation for exact solutions over the interval $\left[0^{\circ}, 360^{\circ}\right)$ In Exercises $83-90$, give all solutions. If necessary, express solutions to the nearest tenth of a degree.
$$4 \cos 2 \theta=8 \sin \theta \cos \theta$$

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Key Concepts

Exact Solutions

Exact solutions refer to answers expressed in exact terms—using fractions, radicals, or known angle measures—rather than decimal approximations. This concept is crucial in trigonometry when the problem requires solutions in a precise algebraic form over a given interval.

Double Angle Identities

Double angle identities are formulas that express trigonometric functions of double angles (like 2?) in terms of functions of a single angle. These identities, such as cos 2? = cos^2? - sin^2? or its alternate forms, are essential for rewriting and simplifying equations that involve trigonometric expressions with multiple angles.

Factoring in Trigonometric Equations

Factoring is a fundamental algebraic technique that involves expressing a mathematical expression as a product of simpler factors. In the context of trigonometric equations, factoring allows one to isolate terms involving trigonometric functions, making it easier to solve the equations by setting each factor equal to zero.

Periodic Nature of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values at regular intervals. When solving equations involving these functions over a specific interval, it is important to consider all angles within that range that satisfy the equation, acknowledging the function’s recurring behavior.

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