\( \operatorname { Sin } 780 \)

To evaluate \( \operatorname{Sin}\,780 \), we need to determine whether the angle is measured in **degrees** or **radians**. Since the notation doesn't specify, I'll provide calculations for both cases. ### 1. Sine of 780 Degrees The sine function has a period of 360 degrees, meaning: \[ \sin(\theta) = \sin(\theta + 360^\circ \times n) \] where \( n \) is any integer. To simplify \( \sin(780^\circ) \): \[ 780^\circ - 720^\circ = 60^\circ \] So, \[ \sin(780^\circ) = \sin(60^\circ) \] And we know: \[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \] **Result:** \[ \sin(780^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \] ### 2. Sine of 780 Radians If the angle is in radians, we'll use the periodicity of the sine function, which has a period of \( 2\pi \) radians: \[ \sin(\theta) = \sin(\theta + 2\pi \times n) \] First, find the equivalent angle between \( 0 \) and \( 2\pi \): \[ 780 \div 2\pi \approx 124.055 \] This means: \[ 780 \text{ radians} = 124 \times 2\pi + \text{remainder} \] Calculate the remainder: \[ 780 - (124 \times 2\pi) \approx 780 - 124 \times 6.28319 \approx 780 - 779.11496 \approx 0.88504 \text{ radians} \] So, \[ \sin(780 \text{ radians}) = \sin(0.88504 \text{ radians}) \] Using a calculator: \[ \sin(0.88504) \approx 0.7741 \] **Result:** \[ \sin(780 \text{ radians}) \approx 0.7741 \] ### Summary - **If 780 is in degrees:** \[ \sin(780^\circ) = \frac{\sqrt{3}}{2} \approx 0.8660 \] - **If 780 is in radians:** \[ \sin(780 \text{ radians}) \approx 0.7741 \] Please confirm the unit of the angle to ensure the correct evaluation.