what is the reference angle of $-768^{circ}$?\nhow many negative sides does your triangle have?\nwhat is the…

Explanation:

Step1: Find a coterminal angle

To find a coterminal angle of (-768^{\circ}), we add multiples of (360^{\circ}). Let's first find (n) such that (-768 + 360n>0). If (n = 3), then (\theta=-768^{\circ}+3\times360^{\circ}=-768^{\circ}+1080^{\circ} = 312^{\circ})

Step2: Determine the reference angle

Since (312^{\circ}) lies in the fourth quadrant ((270^{\circ}<312^{\circ}<360^{\circ})), the formula for the reference angle (\alpha) of an angle (\theta) in the fourth quadrant is (\alpha = 360^{\circ}-\theta) So, (\alpha=360^{\circ}-312^{\circ}=48^{\circ})

Step3: Answer the second question

In a triangle, side lengths are non - negative (by the definition of length in geometry). So the number of negative sides is (0)

Step4: Calculate (\sin(-768^{\circ}))

Since (\sin(-768^{\circ})=\sin(312^{\circ})) (because they are coterminal) and (\sin(312^{\circ})=-\sin(48^{\circ})) (using the identity (\sin(360^{\circ}-\beta)=-\sin\beta) for (\beta = 48^{\circ})) (\sin(48^{\circ})\approx0.7431), so (\sin(-768^{\circ})\approx - 0.7431)

Answer:

The reference angle of (-768^{\circ}) is (48^{\circ}). The number of negative sides of the triangle is (0). (\sin(-768^{\circ})\approx - 0.7431)