Questions: Which of the following values best approximates the length of c in triangle ABC where C=90°, b=12, and a=15° ? C* 3.1058 C ≈ 12.4233 C* 44.7846 C* 46.3644

Solution Steps

To find the length of \( c \) in triangle \( ABC \) where \( C = 90^\circ \), \( b = 12 \), and \( \angle A = 15^\circ \), we can use the cosine function. The cosine of an angle in a right triangle is defined as:

\[ \cos(\angle A) = \frac{b}{c} \]

Using the cosine subtraction formula, we can express \( \cos(15^\circ) \) as:

\[ \cos(15^{\circ}) = \cos(45^{\circ}-30^{\circ}) = \cos(45^{\circ})\cos(30^{\circ}) + \sin(45^{\circ})\sin(30^{\circ}) \]

Substituting the known values:

\[ \cos(45^{\circ}) = \frac{\sqrt{2}}{2}, \quad \cos(30^{\circ}) = \frac{\sqrt{3}}{2}, \quad \sin(45^{\circ}) = \frac{\sqrt{2}}{2}, \quad \sin(30^{\circ}) = \frac{1}{2} \]

Thus, we have:

\[ \cos(15^{\circ}) = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right) = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \]

Now, we can rearrange the cosine formula to find \( c \):

\[ c = \frac{b}{\cos(15^{\circ})} \]

Substituting \( b = 12 \):

\[ c = \frac{12}{\frac{\sqrt{6} + \sqrt{2}}{4}} = 12 \cdot \frac{4}{\sqrt{6} + \sqrt{2}} = \frac{48}{\sqrt{6} + \sqrt{2}} \]

Calculating this gives us approximately:

\[ c \approx 12.4233 \]

Final Answer