sin t and cos t are given. use identities to find the indicated value. where necessary, rationalize…

sin t and cos t are given. use identities to find the indicated value. where necessary, rationalize denominators.\n74) \\( \\sin t = \\frac { 3 } { 7 }, \\cos t = \\frac { 2 \\sqrt { 10 } } { 7 } \\). find \\( \\csc t \\).\na) \\( \\frac { 3 \\sqrt { 10 } } { 20 } \\) b) \\( \\frac { 2 \\sqrt { 10 } } { 3 } \\) c) \\( \\frac { 7 } { 3 } \\) d) \\( \\frac { 7 \\sqrt { 10 } } { 20 } \\)

sin t and cos t are given. use identities to find the indicated value. where necessary, rationalize denominators.\n74) \\( \\sin t = \\frac { 3 } { 7 }, \\cos t = \\frac { 2 \\sqrt { 10 } } { 7 } \\). find \\( \\csc t \\).\na) \\( \\frac { 3 \\sqrt { 10 } } { 20 } \\) b) \\( \\frac { 2 \\sqrt { 10 } } { 3 } \\) c) \\( \\frac { 7 } { 3 } \\) d) \\( \\frac { 7 \\sqrt { 10 } } { 20 } \\)

Answer

Explanation:

Step1: Recall the reciprocal identity

The reciprocal identity for cosecant is $\csc t=\frac{1}{\sin t}$.

Step2: Substitute the given value of $\sin t$

Given $\sin t = \frac{3}{7}$, substituting into the identity gives $\csc t=\frac{1}{\frac{3}{7}}$.

Step3: Simplify the expression

Using the rule for dividing by a fraction ($\frac{1}{\frac{a}{b}}=\frac{b}{a}$), we have $\csc t=\frac{7}{3}$.

Answer:

C. $\frac{7}{3}$