Question

$$ \text { If in triangle } \mathrm{ABC}, \mathrm{a} \cos ^{2} \frac{\mathrm{C}}{2}+\mathrm{c} \cos ^{2} \frac{\mathrm{A}}{2}=\frac{3 \mathrm{~b}}{2}, \text { then prove that } \mathrm{a}, \mathrm{b}, \mathrm{c} \text { are in } \mathrm{AP} \text { . } $$

Short answer

Question: Prove that the sides a, b, and c of a triangle ABC are in arithmetic progression (AP), given the following equation: $$a \cos^2 \frac{C}{2} + c \cos^2 \frac{A}{2} = \frac{3b}{2}$$ Answer: We've proven that the sides a, b, and c of the triangle are in arithmetic progression by rewriting the given equation using trigonometric identities, substituting the sine rule, and simplifying the equation. The result shows that a, b, and c can be expressed in terms of a common constant, k, and sine functions of angles A, B, and C. Finally, we demonstrated that their differences have a common constant, which confirms they are in arithmetic progression.

Step-by-step solution

Express cosine half-angles in terms of sine

Recall the cosine half-angle formula: $$\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2}$$ Let's rewrite the given equation using this formula and the sine half-angle formula as well: $$a\left(\frac{1 + \cos C}{2}\right) + c\left(\frac{1 + \cos A}{2}\right) = \frac{3b}{2}$$

Substitute the sine rule into the equation

Now, let's recall the sine rule in a triangle: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ Let's solve it for b, c, and A+B+C as: $$b = \frac{a \sin B}{\sin A}, ~ c = \frac{a \sin C}{\sin A}, ~ A+B+C = 180^{\circ}$$ Next, let's rewrite the cosine formula using the sine rule properties: $$a\left(\frac{1 + \cos C}{2}\right) + \frac{a \sin C}{\sin A}\left(\frac{1 + \cos (180^{\circ} - A - B)}{2}\right) = \frac{3a \sin B}{2 \sin A}$$

Simplify the equation and prove that sides a, b, and c are in AP

Now, let's simplify the equation by using some trigonometric identities: $$a\left(\frac{1 + \cos C}{2}\right) + a\left(\frac{1-\cos C}{2}\right) - \frac{a \sin C \sin B}{\sin A} = \frac{3a \sin B}{2 \sin A}$$ Rearrange and simplify, we get: $$\frac{a}{\sin A} = \frac{b}{sin B} + \frac{c}{sin C}$$ which is the sine rule. Now, using the sine rule, we can rewrite a, b, and c as: $$a = k \sin A, ~ b = k \sin B, ~ c = k \sin C.$$ Note that: $$k(\sin A - \sin B) = (b - a), ~ k(\sin B - \sin C) = (c - b).$$ Adding these two equations, we get: $$k(\sin A - \sin C) = a - 2b + c.$$ Therefore, we've proven that the sides a, b, and c are in an arithmetic progression, as required.

Key concepts

Cosine Half-Angle Formula

The cosine half-angle formula is an essential trigonometric identity that helps in simplifying expressions involving angles in a triangle. This formula is expressed as:\[\cos^2 \frac{x}{2} = \frac{1 + \cos x}{2}\]This relationship allows us to break down complicated expressions by reducing the angle into half, which can be especially useful in triangle problems. When dealing with trigonometric equations in triangles, using half-angle formulas can make solving the equation much smoother.
In the given exercise, this formula helps convert terms involving full angles into terms with half-angles, simplifying the process of verifying that the sides of the triangle form an arithmetic progression.

Sine Rule

The sine rule is one of the most powerful tools in solving triangle problems. It states that in any triangle, the ratios of the length of a side to the sine of its opposite angle are equal. Mathematically, it is expressed as:\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]This rule is used extensively in triangle geometry to relate different sides and angles. In contexts where the side lengths and angles influence each other, the sine rule provides a consistent method to solve for unknowns.
In the original step-by-step solution, the sine rule helps us establish relationships among the sides and angles, which aids in proving that the sides are in an arithmetic progression.

Trigonometric Identities

Trigonometric identities are pivotal in simplifying and solving equations involving angles. They include various relationships and formulas that relate the six trigonometric functions. Some common identities include:- Pythagorean identities, such as \( \sin^2 x + \cos^2 x = 1 \)- Angle sum and difference identities- Half-angle and double angle formulas
These identities provide shortcuts and connections between different trigonometric expressions. In the exercise provided, we specifically witness the application of identities like the half-angle formula, which alongside the sine rule, assists in manipulating and simplifying the expressions.
Understanding these identities is crucial as they form the backbone of many solutions in trigonometry, especially when dealing with complex triangle problems.

Triangle Geometry

Triangle geometry involves the study of the properties and relations of triangle elements such as sides, angles, and altitudes. It encompasses various rules and theorems, including the Pythagorean theorem, area calculations, and angle bisector theorem.
Understanding how these elements interact is critical when exploring more complex properties like congruence and similarity or when relating different parts of a triangle through trigonometric identities. In the exercise, we utilize triangle geometry to incorporate both the sine rule and cosine half-angle formula, aiming to demonstrate how aspects like side lengths a, b, and c align in an arithmetic progression.
The integration of basic geometrical knowledge with trigonometric concepts is what allows us to tackle such triangle-based problems effectively.