Question

suppose abc is a right triangle with sides a, b, and c and right angle at c. use the pythagorean theorem to find the unknown side length. then find the values of the six trigonometric functions for angle b. rationalize the denominators when applicable. a = 7, c = 9 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.) sec 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.) cot b = (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find side length b using Pythagorean theorem

By the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\), so \(b = \sqrt{c^{2}-a^{2}}\). Substituting \(a = 7\) and \(c = 9\), we get \(b=\sqrt{9^{2}-7^{2}}=\sqrt{81 - 49}=\sqrt{32}=4\sqrt{2}\).

Step2: Define trigonometric - function formulas

\(\sin B=\frac{b}{c}\), \(\cos B=\frac{a}{c}\), \(\tan B=\frac{b}{a}\), \(\sec B=\frac{c}{a}\), \(\csc B=\frac{c}{b}\), \(\cot B=\frac{a}{b}\).

Step3: Calculate \(\sin B\)

\(\sin B=\frac{b}{c}=\frac{4\sqrt{2}}{9}\).

Step4: Calculate \(\cos B\)

\(\cos B=\frac{a}{c}=\frac{7}{9}\).

Step5: Calculate \(\tan B\)

\(\tan B=\frac{b}{a}=\frac{4\sqrt{2}}{7}\).

Step6: Calculate \(\sec B\)

\(\sec B=\frac{c}{a}=\frac{9}{7}\).

Step7: Calculate \(\csc B\)

\(\csc B=\frac{c}{b}=\frac{9}{4\sqrt{2}}=\frac{9\sqrt{2}}{8}\).

Step8: Calculate \(\cot B\)

\(\cot B=\frac{a}{b}=\frac{7}{4\sqrt{2}}=\frac{7\sqrt{2}}{8}\).

Answer:

The unknown side length \(b\) is \(4\sqrt{2}\)
\(\sin B=\frac{4\sqrt{2}}{9}\)
\(\cos B=\frac{7}{9}\)
\(\tan B=\frac{4\sqrt{2}}{7}\)
\(\sec B=\frac{9}{7}\)
\(\csc B=\frac{9\sqrt{2}}{8}\)
\(\cot B=\frac{7\sqrt{2}}{8}\)