Question

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a right triangle has side lengths 7, 24, and 25 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 \(\angle B\):

• Adjacent side \(= 7\)
• Opposite side \(= 24\)
• Hypotenuse \(= 25\)

Step2: Calculate \(\cos B\)

\(\cos B=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{7}{25}\)

Step3: Calculate \(\tan B\)

\(\tan B=\frac{\text{opposite}}{\text{adjacent}}=\frac{24}{7}\)

Step4: Calculate \(\sin B\)

\(\sin B=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{24}{25}\)

Answer:

\(\cos B=\frac{7}{25}\), \(\tan B=\frac{24}{7}\), \(\sin B=\frac{24}{25}\)