Question

$$ \text { In a triangle } \mathrm{ABC}, \mathrm{AC}=9 \text { and } \mathrm{AB}=7 \text { and } \cos (\mathrm{B}-\mathrm{C})=\frac{4}{5} \text { . Find the area of the triangle } \mathrm{ABC} \text { . } $$

Short answer

Answer: To find the area of the triangle, follow these steps: 1. Use the Law of Cosines to find cosB and cosC. 2. Calculate sinB and sinC using the Pythagorean identity. 3. Substitute the values of cosB, cosC, sinB, and sinC in the given trigonometric identity, and solve for sinC. 4. Calculate the area of the triangle using the formula Area = (1/2) * AC * AB * sinC.

Step-by-step solution

Use the Law of Cosines to find cosB and cosC

To find cosB and cosC, we can use the Law of Cosines formula: BC^2 = AC^2 + AB^2 - 2 * AC* AB * cosB (1) BC^2 = AB^2 + AC^2 - 2 * AB* AC * cosC (2) We have AC = 9 and AB = 7. Using these values in equations (1) and (2), we get two equations: BC^2 = 9^2 + 7^2 - 2 * 9 * 7 * cosB BC^2 = 7^2 + 9^2 - 2 * 7 * 9 * cosC Solve these equations to find cosB and cosC.

Calculate sinB and sinC using Pythagorean identity

We know cosB and cosC from Step 1, so we can use the Pythagorean identity to find sinB and sinC: sinB = sqrt(1 - cosB^2) sinC = sqrt(1 - cosC^2)

Substitute in the trigonometric identity

We are given that cos(B-C) = 4/5, and we know the values of cosB, cosC, sinB, and sinC. Now we can substitute these values in the trigonometric identity: 4/5 = cosB*cosC + sinB*sinC Now we can solve for sinC.

Calculate the area of the triangle

Now that we have the value of sinC, we can use the area formula: Area = (1/2) * AC * AB * sinC Substitute AC=9, AB=7, and the value of sinC to calculate the area of the triangle ABC.

Key concepts

Law of Cosines

The Law of Cosines is a crucial tool in trigonometry, especially for solving problems involving triangles. It becomes particularly useful when you need to find unknown sides or angles in any triangle, not just right-angled ones. The formula looks somewhat like the Pythagorean theorem but with an extra term to account for the non-right angle. Here's how it works:
- Consider a triangle with sides labeled as \(a\), \(b\), and \(c\), and the angles opposite these sides are \(A\), \(B\), and \(C\) respectively.- The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \] This formula helps to find the length of a side when you know the lengths of the other two sides and the measure of the included angle.
In our problem, we utilized this law to find expressions for \( \cos B \) and \( \cos C \). By plugging in known values for sides \( AC \) and \( AB \), we derived specific equations that allowed us to express the side \( BC \) in terms of \( \cos B \) and \( \cos C \). Once you understand this process, you can solve for other unknowns in various triangle scenarios effectively.

Trigonometric Identities

Trigonometric identities are like magical tools in the world of mathematics, connecting different trigonometric functions to solve complex problems. One fundamental identity is the Pythagorean identity:
- \( \sin^2(\theta) + \cos^2(\theta) = 1 \), which holds true for any angle \( \theta \).
In our problem, after deriving expressions for \( \cos B \) and \( \cos C \) using the Law of Cosines, we apply this Pythagorean identity to find \( \sin B \) and \( \sin C \):
- \( \sin B = \sqrt{1 - \cos^2 B} \)- \( \sin C = \sqrt{1 - \cos^2 C} \)
These calculations are integral to using trigonometric identities to solve equations involving angles, such as the given identity \( \cos(B-C) = \frac{4}{5} \).By substituting known values of \( \cos B \) and \( \cos C \) along with the Pythagorean identity-derived values of \( \sin B \) and \( \sin C \), we solve the equation for yet another unknown, \( \sin C \). Trigonometric identities thereby transform a seemingly impossible situation into a step-by-step solution.

Triangle Area Calculation

Once you've got the necessary trigonometric values, calculating the area of a triangle becomes straightforward. The formula for the area of a triangle can vary depending on the information available. In trigonometry, when two sides and the included angle are known, the formula is:
- \( \text{Area} = \frac{1}{2}ab\sin(C) \) where \( a \) and \( b \) are two sides, and \( C \) is the included angle.
In the context of our problem, after determining \( \sin C \) through the trigonometric identity and given values, the area of triangle ABC is calculated using the above formula:
- Substitute \( AC = 9 \), \( AB = 7 \), and the calculated \( \sin C \).The formula \( \frac{1}{2} \times 9 \times 7 \times \sin C \) simplifies the process of finding the area and highlights how vital it is to understand each component of the problem. This formula underscores the practical application of trigonometry in determining areas without directly measuring every element of the triangle.