Question

part 5 of 6
write the ratios for sin m, cos m, and tan m. give the exact value and a four - decimal approximation.
type the decimal approximation of the answer rounded to four decimal places.
sin m = 0.3122
(round the final answer to four decimal places as needed)
cos m = (\frac{19}{20})
(type an exact answer in simplified form. type an integer or a fraction.)
cos m = 0.9500
(round the final answer to four decimal places as needed)
tan m =
(type an exact answer in simplified form. type an integer or a fraction.)
help me solve this view an example get more help
right triangle diagram with right angle at k, side mk = 19, side ml = 20, side lk=(sqrt{19})

Explanation:

Step1: Recall the Pythagorean theorem to find the missing side (if needed) and the definition of tangent in a right triangle.

In a right triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For angle \(M\), the opposite side to \(M\) is \(KL = \sqrt{19}\) and the adjacent side is \(KM = 19\).

Step2: Calculate \(\tan M\) using the formula.

\(\tan M=\frac{\text{opposite}}{\text{adjacent}}=\frac{\sqrt{19}}{19}\)
Simplify \(\frac{\sqrt{19}}{19}\), we can also rationalize or just compute the decimal approximation. First, find the decimal value of \(\sqrt{19}\approx 4.3589\). Then \(\tan M=\frac{4.3589}{19}\approx 0.2294\) (rounded to four decimal places). And the exact value is \(\frac{\sqrt{19}}{19}\) or we can simplify \(\frac{\sqrt{19}}{19}=\frac{1}{\sqrt{19}}\) (rationalized form is \(\frac{\sqrt{19}}{19}\)).

Answer:

Exact value: \(\frac{\sqrt{19}}{19}\) (or \(\frac{1}{\sqrt{19}}\) but \(\frac{\sqrt{19}}{19}\) is the simplified exact form with rational denominator), Decimal approximation: \(0.2294\)