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 = 1/6, cos t = √35/6
tan t =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
csc t =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
sec t =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)
cot t =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression. rationalize all denominators.)

Explanation:

Step1: Recall tangent identity

$\tan t=\frac{\sin t}{\cos t}$
Substitute $\sin t = \frac{1}{6}$ and $\cos t=\frac{\sqrt{35}}{6}$:
$\tan t=\frac{\frac{1}{6}}{\frac{\sqrt{35}}{6}}=\frac{1}{\sqrt{35}}=\frac{\sqrt{35}}{35}$

Step2: Recall cosecant identity

$\csc t=\frac{1}{\sin t}$
Substitute $\sin t=\frac{1}{6}$:
$\csc t = 6$

Step3: Recall secant identity

$\sec t=\frac{1}{\cos t}$
Substitute $\cos t=\frac{\sqrt{35}}{6}$:
$\sec t=\frac{6}{\sqrt{35}}=\frac{6\sqrt{35}}{35}$

Step4: Recall cotangent identity

$\cot t=\frac{\cos t}{\sin t}$
Substitute $\sin t = \frac{1}{6}$ and $\cos t=\frac{\sqrt{35}}{6}$:
$\cot t=\sqrt{35}$

Answer:

$\tan t=\frac{\sqrt{35}}{35}$
$\csc t = 6$
$\sec t=\frac{6\sqrt{35}}{35}$
$\cot t=\sqrt{35}$