Question

suppose abc is a right triangle with sides a, b, and c and right angle at c. find the unknown side length using the pythagorean theorem and then find the values of the six trigonometric functions for angle b.
a = 8, c = 17
the unknown side length b is .
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
sin b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cos b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
tan b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
csc b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
sec b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cot b =
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Apply Pythagorean theorem

In a right - triangle with right - angle at C, by the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), where \(c\) is the hypotenuse, and \(a\) and \(b\) are the legs. We know \(a = 8\) and \(c = 17\), so \(b=\sqrt{c^{2}-a^{2}}=\sqrt{17^{2}-8^{2}}=\sqrt{(17 + 8)(17 - 8)}=\sqrt{25\times9}=\sqrt{225}=15\).

Step2: Define sine of angle B

\(\sin B=\frac{b}{c}\), substituting \(b = 15\) and \(c = 17\), we get \(\sin B=\frac{15}{17}\).

Step3: Define cosine of angle B

\(\cos B=\frac{a}{c}\), substituting \(a = 8\) and \(c = 17\), we get \(\cos B=\frac{8}{17}\).

Step4: Define tangent of angle B

\(\tan B=\frac{b}{a}\), substituting \(b = 15\) and \(a = 8\), we get \(\tan B=\frac{15}{8}\).

Step5: Define cosecant of angle B

\(\csc B=\frac{c}{b}\), substituting \(c = 17\) and \(b = 15\), we get \(\csc B=\frac{17}{15}\).

Step6: Define secant of angle B

\(\sec B=\frac{c}{a}\), substituting \(c = 17\) and \(a = 8\), we get \(\sec B=\frac{17}{8}\).

Step7: Define cotangent of angle B

\(\cot B=\frac{a}{b}\), substituting \(a = 8\) and \(b = 15\), we get \(\cot B=\frac{8}{15}\).

Answer:

The unknown side length \(b\) is \(15\).
\(\sin B=\frac{15}{17}\)
\(\cos B=\frac{8}{17}\)
\(\tan B=\frac{15}{8}\)
\(\csc B=\frac{17}{15}\)
\(\sec B=\frac{17}{8}\)
\(\cot B=\frac{8}{15}\)