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{3}{7}, \cos t = \frac{2\sqrt{10}}{7}

tan t = \frac{3\sqrt{10}}{20}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
csc t = \square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Find $\csc t$ via reciprocal identity

$\csc t = \frac{1}{\sin t}$

Step2: Substitute $\sin t = \frac{3}{7}$

$\csc t = \frac{1}{\frac{3}{7}} = \frac{7}{3}$

Step3: Find $\sec t$ via reciprocal identity

$\sec t = \frac{1}{\cos t}$

Step4: Substitute $\cos t = \frac{2\sqrt{10}}{7}$

$\sec t = \frac{1}{\frac{2\sqrt{10}}{7}} = \frac{7}{2\sqrt{10}}$

Step5: Rationalize $\sec t$ denominator

$\sec t = \frac{7}{2\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{7\sqrt{10}}{20}$

Step6: Find $\cot t$ via reciprocal identity

$\cot t = \frac{1}{\tan t}$

Step7: Substitute $\tan t = \frac{3\sqrt{10}}{20}$

$\cot t = \frac{1}{\frac{3\sqrt{10}}{20}} = \frac{20}{3\sqrt{10}}$

Step8: Rationalize $\cot t$ denominator

$\cot t = \frac{20}{3\sqrt{10}} \times \frac{\sqrt{10}}{\sqrt{10}} = \frac{20\sqrt{10}}{30} = \frac{2\sqrt{10}}{3}$

Answer:

$\csc t = \frac{7}{3}$
$\sec t = \frac{7\sqrt{10}}{20}$
$\cot t = \frac{2\sqrt{10}}{3}$