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 = 7, c = 25 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 \(c\), \(c^{2}=a^{2}+b^{2}\). We want to find \(b\), so \(b = \sqrt{c^{2}-a^{2}}\).

Step2: Substitute given values

Given \(a = 7\) and \(c = 25\), then \(b=\sqrt{25^{2}-7^{2}}=\sqrt{(25 + 7)(25 - 7)}\) (using the difference - of - squares formula \(x^{2}-y^{2}=(x + y)(x - y)\)).

Step3: Calculate the values inside the square root

\((25 + 7)(25 - 7)=32\times18 = 576\).

Step4: Find the square root

\(b=\sqrt{576}=24\).

Answer:

24