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 = 3, c = 5
the unknown side length b is
(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 sides \(a\), \(b\), and hypotenuse \(c\), \(a^{2}+b^{2}=c^{2}\). We know \(a = 3\) and \(c = 5\), so we can solve for \(b\): \(b=\sqrt{c^{2}-a^{2}}\).

Step2: Substitute values

Substitute \(a = 3\) and \(c = 5\) into the formula: \(b=\sqrt{5^{2}-3^{2}}=\sqrt{25 - 9}=\sqrt{16}\).

Step3: Simplify

\(\sqrt{16}=4\).

Answer:

4