Question

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a right triangle has side lengths 9, 40, and 41 as shown below. use these lengths to find cos b, tan b, and sin b.

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) for an angle \(\theta\). For angle \(B\), the adjacent side is \(BC = 9\), the opposite side is \(AC = 40\), and the hypotenuse is \(AB=41\).

Step2: Calculate \(\cos B\)

Using the formula \(\cos B=\frac{\text{adjacent}}{\text{hypotenuse}}\), we substitute the values: \(\cos B=\frac{BC}{AB}\). So, \(\cos B=\frac{9}{41}\).

Step3: Calculate \(\tan B\)

Using the formula \(\tan B=\frac{\text{opposite}}{\text{adjacent}}\), we substitute the values: \(\tan B=\frac{AC}{BC}\). So, \(\tan B=\frac{40}{9}\).

Step4: Calculate \(\sin B\)

Using the formula \(\sin B=\frac{\text{opposite}}{\text{hypotenuse}}\), we substitute the values: \(\sin B=\frac{AC}{AB}\). So, \(\sin B=\frac{40}{41}\).

Answer:

\(\cos B=\frac{9}{41}\), \(\tan B=\frac{40}{9}\), \(\sin B=\frac{40}{41}\)