Question

12 multiple choice 1 point evaluate the trigonometric function of the given quadrantal angle. sin 8π 1 -1 undefined 0 13 multiple choice 1 point evaluate the trigonometric function of the given quadrantal angle. tan 1440° undefined 1 0

Explanation:

Step1: Use the periodicity of sine function

The sine function \(y = \sin(x)\) has a period of \(2\pi\). That is, \(\sin(x+2k\pi)=\sin(x)\) for any integer \(k\).
For \(y = \sin(8\pi)\), we can write \(8\pi=4\times2\pi\). Let \(x = 0\) and \(k = 4\) in the formula \(\sin(x + 2k\pi)=\sin(x)\). Then \(\sin(8\pi)=\sin(0 + 4\times2\pi)\).
Since \(\sin(0)=0\), we have \(\sin(8\pi)=0\).

Step2: Use the periodicity of tangent function

The tangent function \(y=\tan(x)\) has a period of \(180^{\circ}\) (or \(\pi\) radians). The formula is \(\tan(x + 180^{\circ}k)=\tan(x)\) for any integer \(k\).
For \(y=\tan(1440^{\circ})\), we first find \(k\) such that \(x+180^{\circ}k = 1440^{\circ}\). We know that \(1440\div180 = 8\). Let \(x = 0^{\circ}\) and \(k = 8\) in the formula \(\tan(x+180^{\circ}k)=\tan(x)\). Then \(\tan(1440^{\circ})=\tan(0^{\circ}+8\times180^{\circ})\).
Since \(\tan(0^{\circ}) = 0\), we have \(\tan(1440^{\circ})=0\).

Answer:

For \(\sin(8\pi)\), the answer is \(0\).
For \(\tan(1440^{\circ})\), the answer is \(0\).