Solve each equation over the interval [0°, 360°). Write solutions as exact values or to the nearest tenth of a degree, as appropriate.
sin 2θ = cos 2θ +1

Verified step by step guidance

1. Start with the given equation: \(\sin 2\theta = \cos 2\theta + 1\).
2. Rearrange the equation to isolate terms on one side: \(\sin 2\theta - \cos 2\theta = 1\).
3. Use the identity for expressing \(\sin A - \cos A\) in a single trigonometric function: recall that \(\sin x - \cos x = \sqrt{2} \sin \left(x - 45^\circ\right)\), so rewrite as \(\sqrt{2} \sin \left(2\theta - 45^\circ\right) = 1\).
4. Divide both sides by \(\sqrt{2}\) to isolate the sine term: \(\sin \left(2\theta - 45^\circ\right) = \frac{1}{\sqrt{2}}\).
5. Solve for the angle inside the sine function by finding all angles in \([0^\circ, 360^\circ)\) where \(\sin \alpha = \frac{1}{\sqrt{2}}\), then solve for \(\theta\) by reversing the substitution: \(2\theta - 45^\circ = \alpha\).

Key Concepts

Double-Angle Identities

Double-angle identities express trigonometric functions of twice an angle in terms of single angles, such as sin(2θ) = 2 sin θ cos θ and cos(2θ) = cos² θ - sin² θ. These identities help simplify or rewrite equations involving 2θ to solve for θ.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle solutions within a given interval. This often requires using identities, algebraic manipulation, and considering the periodic nature of sine and cosine functions.

Interval and Solution Representation

When solving trigonometric equations over [0°, 360°), solutions must be expressed within this range. Answers can be exact values (like π/4) or decimal approximations rounded appropriately, ensuring all valid solutions in the interval are included.