Question

If in a \(\triangle \mathrm{ABC}, \operatorname{acos}^{2}\left(\frac{\mathrm{C}}{2}\right)+\mathrm{c} \cos ^{2}\left(\frac{\mathrm{A}}{2}\right)=\frac{3 \mathrm{~b}}{2}\), then the sides \(a, c, b\) (1) are in A.P. (2) are in G.P. (3) are in H.P. (4) satisfy \(a+c=2 b\)

Short answer

The sides satisfy: \(a + c = 2b\) (Option 4)

Step-by-step solution

Simplify the given equation using cosine half-angle identities

Use the cosine half-angle identities: \[ \text{cos}^2 \left(\frac{C}{2}\right) = \frac{1 + \text{cos}C}{2} \] and \[ \text{cos}^2 \left(\frac{A}{2}\right) = \frac{1 + \text{cos}A}{2} \] Substitute these into the given equation: \[ a \text{cos}^2 \left(\frac{C}{2}\right) + c \text{cos}^2 \left(\frac{A}{2}\right) = \frac{3b}{2} \] becomes \[ a \left(\frac{1 + \text{cos}C}{2}\right) + c \left(\frac{1 + \text{cos}A}{2}\right) = \frac{3b}{2} \]

Expand the equation

Multiply through by 2 to clear the fraction: \[ a(1 + \text{cos}C) + c(1 + \text{cos}A) = 3b \]Expanding the terms yields: \[ a + a \text{cos}C + c + c \text{cos}A = 3b \]

Substitute cosine laws

Utilize the cosine law for a triangle: \[ \text{cos}C = \frac{a^2 + b^2 - c^2}{2ab} \] and \[ \text{cos}A = \frac{b^2 + c^2 - a^2}{2bc} \] Substitute these into the expanded equation: \[ a + c + a \left(\frac{a^2 + b^2 - c^2}{2ab}\right) + c \left(\frac{b^2 + c^2 - a^2}{2bc}\right) = 3b \]

Simplify using algebraic manipulation

Combine and simplify the equation step-by-step, focusing on clearing fractions and combining like terms. This process will show that terms involving \(a\),\(b\), and \(c\) must satisfy the condition for (1), (2), (3), or (4). After simplifications, it becomes evident that: \[ a + c = 2b \]

Key concepts

cosine half-angle identities

Let's first understand what cosine half-angle identities are. These identities are used to express cosine functions in a simpler form, particularly when dealing with half-angles. The relevant identities are: \[ \text{cos}^2 \left(\frac{C}{2}\right) = \frac{1 + \text{cos}C}{2} \] \ and \[ \text{cos}^2 \left(\frac{A}{2}\right) = \frac{1 + \text{cos}A}{2} \] Substituting these identities into a given equation helps simplify the problem. This step is often crucial in problems involving trigonometric identities, as it breaks down more complex expressions into manageable parts. In our exercise, using these identities transforms the original equation, making it possible to further manipulate and solve it.

cosine law

The cosine law, also known as the Law of Cosines, is essential for solving many trigonometric problems. It relates the sides of a triangle to the cosine of one of its angles. The law is given by: \[ \text{cos}C = \frac{a^2 + b^2 - c^2}{2ab} \] \ and \[ \text{cos}A = \frac{b^2 + c^2 - a^2}{2bc} \] In our problem, we substitute these expressions into the simplified equation obtained earlier. This substitution allows us to connect the angles and sides of the triangle directly. The cosine law is particularly useful when we have information about the sides of a triangle and need to find the angles, or vice versa. By linking the lengths of sides with the angles, we can solve for the unknowns in the problem.

algebraic manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to isolate the desired variable or simplify complex expressions. In our context, it involves performing operations such as multiplying through by a common factor, combining like terms, and eliminating fractions. After applying the cosine half-angle identities and substituting the cosine law into our equation, we perform algebraic manipulations to simplify the resulting equation: \[ a + a \text{cos}C + c + c \text{cos}A = 3b \] \ becomes \[ a + c + a \left(\frac{a^2 + b^2 - c^2}{2ab}\right) + c \left(\frac{b^2 + c^2 - a^2}{2bc}\right) = 3b \] By carefully combining and simplifying, we eventually derive the relationship between the sides of the triangle: \[ a + c = 2b \] Mastering algebraic manipulation is key in solving complex trigonometric problems like this one. It requires attention to detail and a solid understanding of algebraic principles.