Question

question 6 of 10
what does the law of cosines reduce to when dealing with a right triangle?
a. the formula for a triangles area
b. the law of cosines cannot be applied to a right triangle.
c. the pythagorean theorem
d. the law of sines

Explanation:

The law of cosines is given by \( c^{2}=a^{2}+b^{2}-2ab\cos C \), where \( C \) is the angle opposite side \( c \) in a triangle. In a right triangle, one of the angles (say \( C \)) is \( 90^{\circ} \), and \( \cos(90^{\circ}) = 0 \). Substituting \( \cos C=0 \) into the law of cosines formula, we get \( c^{2}=a^{2}+b^{2}-2ab\times0=a^{2}+b^{2} \), which is the Pythagorean theorem. Option A is incorrect because the law of cosines relates to side lengths, not area. Option B is incorrect as the law of cosines can be applied to right triangles (and it reduces to the Pythagorean theorem). Option D is incorrect as the law of sines is a different relationship (\( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \)) and not related to the reduction of the law of cosines in a right triangle.

Answer:

C. The Pythagorean theorem