Question

1. sin(-t) = \frac{3}{8}

a. sin t
b. csc t

Explanation:

Part a: Find \(\sin t\)

Step1: Recall the odd function property of sine

The sine function is an odd function, which means \(\sin(-x)=-\sin(x)\) for any real number \(x\).
Given \(\sin(-t)=\frac{3}{8}\), by the odd function property of sine, we have \(\sin(-t)=-\sin(t)\).

Step2: Solve for \(\sin t\)

Substitute \(\sin(-t)=\frac{3}{8}\) into \(\sin(-t)=-\sin(t)\), we get \(\frac{3}{8}=-\sin(t)\).
Multiply both sides by \(- 1\) to solve for \(\sin(t)\): \(\sin(t)=-\frac{3}{8}\).

Part b: Find \(\csc t\)

Step1: Recall the reciprocal identity of cosecant

The cosecant function is the reciprocal of the sine function, that is \(\csc(x)=\frac{1}{\sin(x)}\) for \(\sin(x)
eq0\).
We already found that \(\sin(t)=-\frac{3}{8}\) from part (a).

Step2: Calculate \(\csc t\)

Substitute \(\sin(t)=-\frac{3}{8}\) into the reciprocal identity \(\csc(t)=\frac{1}{\sin(t)}\), we get \(\csc(t)=\frac{1}{-\frac{3}{8}}\).
Simplify the right - hand side: \(\csc(t)=-\frac{8}{3}\).

Part a Answer: \(\sin t = \boldsymbol{-\frac{3}{8}}\)

Part b Answer: \(\csc t=\boldsymbol{-\frac{8}{3}}\)

Answer:

Step1: Recall the reciprocal identity of cosecant

The cosecant function is the reciprocal of the sine function, that is \(\csc(x)=\frac{1}{\sin(x)}\) for \(\sin(x)
eq0\).
We already found that \(\sin(t)=-\frac{3}{8}\) from part (a).

Step2: Calculate \(\csc t\)

Substitute \(\sin(t)=-\frac{3}{8}\) into the reciprocal identity \(\csc(t)=\frac{1}{\sin(t)}\), we get \(\csc(t)=\frac{1}{-\frac{3}{8}}\).
Simplify the right - hand side: \(\csc(t)=-\frac{8}{3}\).

Part a Answer: \(\sin t = \boldsymbol{-\frac{3}{8}}\)

Part b Answer: \(\csc t=\boldsymbol{-\frac{8}{3}}\)