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 = 1/4, cos t = √15/4
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 the tangent formula

$\tan t=\frac{\sin t}{\cos t}$
Substitute $\sin t = \frac{1}{4}$ and $\cos t=\frac{\sqrt{15}}{4}$ into the formula:
$\tan t=\frac{\frac{1}{4}}{\frac{\sqrt{15}}{4}}=\frac{1}{\sqrt{15}}=\frac{\sqrt{15}}{15}$

Step2: Recall the cosecant formula

$\csc t=\frac{1}{\sin t}$
Substitute $\sin t=\frac{1}{4}$ into the formula:
$\csc t = 4$

Step3: Recall the secant formula

$\sec t=\frac{1}{\cos t}$
Substitute $\cos t=\frac{\sqrt{15}}{4}$ into the formula:
$\sec t=\frac{4}{\sqrt{15}}=\frac{4\sqrt{15}}{15}$

Step4: Recall the cotangent formula

$\cot t=\frac{\cos t}{\sin t}$
Substitute $\sin t = \frac{1}{4}$ and $\cos t=\frac{\sqrt{15}}{4}$ into the formula:
$\cot t=\sqrt{15}$

Answer:

$\tan t=\frac{\sqrt{15}}{15}$
$\csc t = 4$
$\sec t=\frac{4\sqrt{15}}{15}$
$\cot t=\sqrt{15}$