Question

the angle t is an acute angle and sin t and cos t are given. use identities to find tan t, csc t, sec t, and cot t. where necessary, rationalize denominators.

\\( \sin t = \frac { 5 } { 7 } , \cos t = \frac { 2 \sqrt { 6 } } { 7 } \\)

\\( \tan t = \frac { 5 \sqrt { 6 } } { 12 } \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

\\( \csc t = \frac { 7 } { 5 } \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

\\( \sec t = \square \\)
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall the reciprocal identity for secant

The reciprocal identity for secant is \(\sec t=\frac{1}{\cos t}\).

Step2: Substitute the given value of \(\cos t\)

We know that \(\cos t = \frac{2\sqrt{6}}{7}\). Substituting this into the identity for \(\sec t\), we get \(\sec t=\frac{1}{\frac{2\sqrt{6}}{7}}\).

Step3: Simplify the expression

To simplify \(\frac{1}{\frac{2\sqrt{6}}{7}}\), we multiply the numerator and denominator by the reciprocal of the denominator, which is \(\frac{7}{2\sqrt{6}}\). So, \(\sec t = 1\times\frac{7}{2\sqrt{6}}=\frac{7}{2\sqrt{6}}\).

Step4: Rationalize the denominator

To rationalize the denominator, we multiply the numerator and denominator by \(\sqrt{6}\). So, \(\sec t=\frac{7\times\sqrt{6}}{2\sqrt{6}\times\sqrt{6}}=\frac{7\sqrt{6}}{2\times6}=\frac{7\sqrt{6}}{12}\).

Answer:

\(\frac{7\sqrt{6}}{12}\)