Given that \( \cos \theta \frac{8}{17} \cdot \) Cslculat \( \operatorname{cosec} \theta \) if \( \cos \theta=\operatorname{cin} 80^{\circ} \), find \( \theta \)

There appears to be some confusion in the way the question is printed. One reading is that there are two parts: 1. Given that   cos θ = 8⁄17, calculate cosec θ. 2. Given that   cos θ = cos 80°, find all possible values of θ (or at least the principal one). I will explain both parts. ────────────────────────────── Part 1. Finding cosec θ when cos θ = 8⁄17 Step 1. Use the Pythagorean identity:   sin²θ + cos²θ = 1. Since cos θ = 8⁄17, we have   sin²θ = 1 − (8⁄17)² = 1 − 64⁄289 = (289 − 64)/289 = 225/289. Taking a (positive) square root (if we assume the angle is in the first quadrant),   sin θ = 15⁄17. Step 2. Recall that the cosecant is the reciprocal of the sine:   cosec θ = 1/sin θ = 17/15. ────────────────────────────── Part 2. Finding θ when cos θ = cos 80° The equation   cos θ = cos 80° has the general solution (in degrees):   θ = 80° + 360°·k  or  θ = –80° + 360°·k   for any integer k. If one is asked for the principal value (typically in the interval 0° ≤ θ < 360°), then the solutions are   θ = 80°  and  θ = 360° – 80° = 280°. ────────────────────────────── Thus, the answers are: • If cos θ = 8⁄17 (with θ acute), then sin θ = 15⁄17 and so cosec θ = 17/15. • If cos θ = cos 80°, then, in general, θ = 80° + 360°·k or θ = –80° + 360°·k (for any integer k); in [0°, 360°] these give θ = 80° and 280°. Hopefully, this clarifies the problem.