Questions: Question 1 14 pts Given tan(θ) = 5/6, then sin(θ) = a/b. Determine the values of a and b. a=[ Select ] b=[ Select ]

Solution Steps

We are given that \(\tan(\theta) = \frac{5}{6}\). Recall that \(\tan(\theta)\) is defined as the ratio of the opposite side to the adjacent side in a right triangle. Therefore, we can represent \(\theta\) as an angle in a right triangle where the opposite side is 5 units and the adjacent side is 6 units.

Using the Pythagorean theorem, we can find the hypotenuse \(h\) of the right triangle: \[ h = \sqrt{5^2 + 6^2} = \sqrt{25 + 36} = \sqrt{61}. \]

The sine of an angle in a right triangle is defined as the ratio of the opposite side to the hypotenuse. Therefore: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{5}{\sqrt{61}}. \] To rationalize the denominator, multiply the numerator and denominator by \(\sqrt{61}\): \[ \sin(\theta) = \frac{5\sqrt{61}}{61}. \]

Final Answer