Question

use the pythagorean theorem to find the length of the missing side of the right triangle. then find the value of each of the six trigonometric functions of θ. the length of the missing side of the right triangle is □. (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall Pythagorean Theorem

The Pythagorean Theorem for a right triangle is \(a^{2}+b^{2}=c^{2}\), where \(a\) and \(b\) are the legs, and \(c\) is the hypotenuse. Here, \(a = 6\), \(b=8\), so we substitute these values into the formula: \(6^{2}+8^{2}=c^{2}\).

Step2: Calculate the squares

Calculate \(6^{2}=36\) and \(8^{2} = 64\). Then \(36 + 64=c^{2}\), so \(100=c^{2}\).

Step3: Solve for \(c\)

Take the square root of both sides: \(c=\sqrt{100}=10\) (we take the positive root since length can't be negative).

Answer:

10