Question

which equation can be used to find b in the triangle below?

( \tan b=\frac{8}{10} )
( \tan b=\frac{8}{6} )
( cos b=\frac{10}{6} )
( cos b=\frac{10}{8} )

Explanation:

Step1: Recall trigonometric ratios

In a right - triangle, the tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side, and the cosine of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. For angle \(B\) in the right - triangle \(ABC\) (right - angled at \(A\)):

• The side opposite to angle \(B\) is \(AC = 8\)
• The side adjacent to angle \(B\) is \(AB=6\)
• The hypotenuse is \(BC = 10\)

Step2: Analyze each option

• Option 1: \(\tan B=\frac{8}{10}\)

The formula for tangent is \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\frac{8}{10}\) is \(\frac{\text{opposite}}{\text{hypotenuse}}\), so this is incorrect.

• Option 2: \(\tan B=\frac{8}{6}\)

Since \(\tan B=\frac{\text{opposite to }B}{\text{adjacent to }B}=\frac{AC}{AB}=\frac{8}{6}\), this formula is correct.

• Option 3: \(\cos B=\frac{10}{6}\)

The formula for cosine is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). The hypotenuse is the longest side in a right - triangle, and \(\frac{10}{6}> 1\), but the value of cosine of an angle in a right - triangle is always between \(0\) and \(1\) (since adjacent side length is less than hypotenuse length), so this is incorrect.

• Option 4: \(\cos B=\frac{10}{8}\)

Again, \(\frac{10}{8}>1\) and \(\cos B=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{AB}{BC}=\frac{6}{10}
eq\frac{10}{8}\), so this is incorrect.

Answer:

\(\boldsymbol{\tan B=\frac{8}{6}}\) (the second option: \(\tan B=\frac{8}{6}\))