The law of cosines reduces to the Pythagorean theorem whenever the triangle is acute.
Check the final answer first, then review the worked steps.
Answer
False
Step-by-step solution
- Recall the Law of Cosines: The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides $a$, $b$, and $c$, and with angle $C$ opposite side $c$, the law is stated as: $$c^2 = a^2 + b^2 - 2ab \cos(C)$$
- Recall the Pythagorean Theorem: The Pythagorean Theorem applies to right-angled triangles and states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. For a right-angled triangle with sides $a$, $b$, and hypotenuse $c$, the theorem is: $$c^2 = a^2 + b^2$$
- Analyze the condition for reduction: The Law of Cosines reduces to the Pythagorean Theorem when the term $-2ab \cos(C)$ becomes zero. This happens when $\cos(C) = 0$.
- Determine the angle for $\cos(C) = 0$: The cosine of an angle is zero when the angle is $90^\circ$ (or $\frac{\pi}{2}$ radians). Therefore, $\cos(C) = 0$ when $C = 90^\circ$.
- Interpret the angle: An angle of $90^\circ$ means the triangle is a right-angled triangle. The Pythagorean Theorem is specifically for right-angled triangles.
- Evaluate the statement: The statement claims the Law of Cosines reduces to the Pythagorean Theorem whenever the triangle is acute. An acute triangle is a triangle where all three angles are less than $90^\circ$. If all angles are less than $90^\circ$, then $\cos(C)$ will be positive for all angles $C$ (since $0^\circ < C < 90^\circ$). This means the term $-2ab \cos(C)$ will be negative, not zero. Therefore, the Law of Cosines does not reduce to the Pythagorean Theorem for acute triangles. It reduces to the Pythagorean Theorem for right-angled triangles.