Question

which equation can be solved to find one of the missing side lengths in the triangle? triangle with right angle at c, angle at b is 60°, hypotenuse ab is 12 units, side bc is a, side ac is b. options: \\(\cos(60°) = \frac{a}{12}\\), \\(\cos(60°) = \frac{12}{a}\\), \\(\cos(60°) = \frac{b}{a}\\), \\(\cos(60°) = \frac{12}{b}\\)

Explanation:

Step1: Recall the cosine definition in a right triangle

In a right triangle, the cosine of an acute angle is defined as the ratio of the adjacent side to the hypotenuse. For angle \( B = 60^\circ \) in right triangle \( ABC \) (right - angled at \( C \)):

• The hypotenuse is \( AB = 12 \) units.
• The adjacent side to angle \( B \) is \( BC=a \).
• The formula for cosine of an angle \( \theta \) in a right triangle is \( \cos(\theta)=\frac{\text{Adjacent}}{\text{Hypotenuse}} \).

Step2: Apply the cosine formula to angle \( B = 60^\circ \)

Substituting \( \theta = 60^\circ \), adjacent side \( = a \) and hypotenuse \( = 12 \) into the cosine formula, we get:
\( \cos(60^\circ)=\frac{a}{12} \)

Answer:

\( \cos(60^\circ)=\frac{a}{12} \) (the first option among the given options with this formula)