Find an angle thetaθ coterminal to minus, 177, degrees−177 ∘ , where 0, degrees, is less than or equal to, theta, is less than, 360, degrees0 ∘ ≤θ<360 ∘ .

Step 1: Break Down the Problem

To find a coterminal angle θ with -177°, we need to add or subtract multiples of 360° until we get an angle within the range $0° \leq \theta < 360°$.

Step 2: Relevant Concepts

Coterminal angles are created by adding or subtracting 360° from the given angle. Therefore, the formula we will use is: $\theta = -177° + 360°n$ where $n$ is an integer.

Step 3: Analysis and Detail

1. Start with the given angle: $-177°$
2. To find a positive angle, set $n = 1$: $\theta = -177° + 360°(1) = -177° + 360° = 183°$
3. This angle $183°$ falls within the specified range of $0° \leq \theta < 360°$.

Step 4: Verify and Summarize

We calculated: $\theta = -177° + 360° = 183°$ Since $183°$ is a valid angle within the range, it is indeed coterminal with $-177°$.

Final Answer

The angle θ coterminal to $-177°$ within the range $0° \leq \theta < 360°$ is $\theta = 183°.$