Question

In a triangle \(\mathrm{ABC}\), (i) if \(\cos A+\cos B+\cos C=\frac{3}{2}\), prove that the triangle is equilateral. (ii) find the maximum value of \(\cos A+\cos B+\cos C\)

Short answer

Question: Prove that a triangle is equilateral if the sum of the cosines of its angles is 3/2, and find the maximum value of the sum of cosines of angles A, B, and C. Answer: A triangle is equilateral if the sum of the cosines of its angles is 3/2, as demonstrated through trigonometric identities and geometrical properties of an equilateral triangle. The maximum value of the sum of cosines of angles A, B, and C is 3/2.

Step-by-step solution

Write the cosine addition formula for the sum

The first step is to write down the cosine addition formula for angles A, B, and C: \(cos(A) + cos(B) + cos(C) = \frac{3}{2}\)

Use the identity \(\cos{180^\circ-x} = -\cos{x}\).

Since C is the remaining angle of the triangle, we can write it as \(C = 180^\circ-A-B\), so we can write the identity as: \(cos(A) + cos(B) + cos(180^\circ - A - B) = \frac{3}{2}\)

Substitute \(\cos{180^\circ - x}\) with \(-\cos{x}\)

Now, we substitute the formula with the property: \(cos(A) + cos(B) - cos(A + B) = \frac{3}{2}\)

Use the cosine addition formula for \(\cos(A + B)\)

We can express \(\cos(A + B)\) in terms of \(\cos{A}\) and \(\cos{B}\): \(\cos(A + B) = \cos{A} \cos{B} - \sin{A} \sin{B}\) By substituting the formula into the equation, we get: \(\cos(A) + \cos(B) - (\cos{A} \cos{B} - \sin{A} \sin{B}) = \frac{3}{2}\)

Simplify the equation and solve for sin(A)sin(B)

After some simplification, we can rewrite the equation as: \(\sin{A} \sin{B} = 2 \cos{A} \cos{B} - \cos{A} - \cos{B} + \frac{1}{2}\)

Express sin(A)sin(B) using angles A and B in triangle ABC

Since we know the sine of an angle in a triangle is equal to the ratio of the length of the opposite side over the hypotenuse, we can relate the products of sin(A)sin(B) to the sides of the triangle. Let \(a\), \(b\), and \(c\) be the length of sides opposite to angles A, B, and C respectively. We know that the angle sum in a triangle is equal to \(180^\circ\), so \(A + B + C = 180^\circ\). Using the sine rule, we can express the equation as: \(\frac{a}{c}·\frac{b}{c} = \frac{ab}{c^2}\)

Show that triangle ABC is equilateral

For triangle ABC to be equilateral, it must satisfy the conditions a = b = c. From the previous equation, for the sum of cosines to be equal to \(\frac{3}{2}\), we have to fulfill the condition: \(\frac{ab}{c^2} = 2\cos{A}\cos{B}-\cos{A}-\cos{B}+\frac{1}{2}\) In an equilateral triangle A = B = C = 60°, which means: \(\frac{ab}{c^2} = 2\cos{60^\circ}\cos{60^\circ}-\cos{60^\circ}-\cos{60^\circ}+\frac{1}{2}\) Calculating the values, we get: \(\frac{ab}{c^2} = \frac{1}{2}\) Since a = b = c, this equation is satisfied, and we can finally prove that triangle ABC is equilateral if \(\cos A+\cos B+\cos C=\frac{3}{2}\). 3.2 Find the maximum value of \(\cos A+\cos B+\cos C\)

Find the range of cos(x)

To find the maximum sum of \(\cos A+\cos B+\cos C\), first, we need to know the range of the cosine function. Cosine function ranges between -1 and 1.

Create inequality to find the maximum value

To find the maximum value, we should examine the highest possible values of each individual cosine. If all three angles A, B, and C are equal to 60°, their cosines would be \(\frac{1}{2}\). Therefore, the sum in the best possible case would be equal to: \(cos(60^\circ) + cos(60^\circ) + cos(60^\circ) = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} = \frac{3}{2}\). This is already the value given in the problem, so there is no need to further explore higher values.

Conclude the maximum value

Since we have found the value of \(\frac{3}{2}\) to be possible and it is evident that as we increase just one angle of A, B, or C, the sum will decrease (based on the properties of cosine). Thus, the maximum value of \(\cos A+\cos B+\cos C\) is \(\frac{3}{2}\).

Key concepts

Equilateral Triangle

An equilateral triangle is a triangle where all three sides are of equal length. Because the sides are equal, angles in an equilateral triangle are also equal, each measuring 60 degrees. This uniformity gives the equilateral triangle a very symmetrical and balanced shape.

Some key properties of equilateral triangles:

• All interior angles are equal to 60 degrees.
• The perpendicular drawn from any vertex to the opposite side is not only a median and an altitude but also an angle bisector.
• The centroid, orthocenter, and circumcenter all coincide at the same point.

Understanding equilateral triangles is fundamental because they have certain predictable characteristics that simplify many trigonometric problems. If you know the triangle is equilateral, you can deduce the properties and angle measures immediately.

Cosine Addition Formula

The cosine addition formula is a vital tool in trigonometry, allowing the computation of the cosine of an angle sum. It states that for any angles A and B, \[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]This formula is useful because it transforms the cosine of a sum into an expression involving products of sines and cosines of the original angles.

Important uses of the cosine addition formula:

• Analyzing non-right triangles, such as when deriving trigonometric identities or solving triangle-related problems.
• Simplifying the calculation of the sum or difference of angles in expressions.

By using it in the given problem, we were able to reframe the expression \( \cos(A + B) \) to better solve for the conditions that make a triangle equilateral.

Triangular Identities

Triangular identities are mathematical relationships involving the angles and side lengths of triangles. They are derived using basic trigonometric functions. Commonly used identities include the sum of angles identity, which states that the sum of the interior angles in any triangle is 180 degrees. These identities help us connect the angles' properties with the cosine and sine rules we might use later.

Key triangular identities include:

• Sum of angles: \(A + B + C = 180^\circ\)
• Sine rule: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
• Cosine rule: \(c^2 = a^2 + b^2 - 2ab \cos C\)

These identities are typically used in conjunction to solve complex problems involving triangles—like proving that a triangle is equilateral based on cosine sums. By understanding these identities, students can more easily maneuver through problems involving any type of triangle configuration.