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 a = 6. sinθ = 3/5 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) cosθ = 4/5 (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.) tanθ = □ (simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the definition of tangent

In a right triangle, the tangent of an angle \(\theta\) is defined as the ratio of the length of the opposite side to the length of the adjacent side. So, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Identify opposite and adjacent sides

For angle \(\theta\) in triangle \(ABC\), the opposite side is \(a = 6\) and the adjacent side is \(b = 8\).

Step3: Calculate \(\tan\theta\)

Substitute the values into the formula: \(\tan\theta=\frac{a}{b}=\frac{6}{8}\). Simplify the fraction by dividing numerator and denominator by their greatest common divisor, which is 2. So, \(\frac{6\div2}{8\div2}=\frac{3}{4}\).

Answer:

\(\frac{3}{4}\)