Understanding the Angles of a Right-Angled Triangle

A right-angled triangle is a triangle that contains a right angle (90°). In this type of triangle, the relationship between the angles and sides plays a crucial role in solving geometric problems. Let's explore the angles and how to determine them in a right-angled triangle using the sine and cosine rules.

Introduction to Right-Angled Triangle ABC

Consider a right-angled triangle ABC, where angle B is 90°. The sides are given as follows: AB 3, BC 4, and CA 5. This triangle is a classic example of a Pythagorean triple, confirming that it is indeed a right-angled triangle.

Solving for the Angles Using the Sine Rule

The sine rule states that the ratio of the length of a side to the sine of the opposite angle is constant for all three sides of the triangle. We can use this rule to find angles C and A.

Step 1: Apply the sine rule

Using the sine rule: [frac{AB}{sin C} frac{BC}{sin A} frac{CA}{sin B}]

Since (sin B 1) (for a right angle), we can rewrite the equation as:

[frac{3}{sin C} frac{4}{sin A} frac{5}{sin 90°} 5]

Step 2: Solve for (sin C)

[frac{3}{sin C} 5] [sin C frac{3}{5}]

Step 3: Solve for (sin A)

[frac{4}{sin A} 5] [sin A frac{4}{5}]

To find the angles, we take the inverse sine (arcsin) of the ratios.

[sin C frac{3}{5} implies C sin^{-1}left(frac{3}{5}right)]

[sin A frac{4}{5} implies A sin^{-1}left(frac{4}{5}right)]

Hence, we have the measures of angles A, B, and C.

Solving for the Angles Using the Cosine Rule

The cosine rule can be applied to a different type of problem, where we know the lengths of all sides. The cosine rule states that for a side of a triangle, the square of its length is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle.

Step 1: Apply the cosine rule to find angle A

[a 4, b 5, c 3]

[cos A frac{b^2 c^2 - a^2}{2bc}]

[cos A frac{5^2 3^2 - 4^2}{2 times 5 times 3}] [cos A frac{25 9 - 16}{30} frac{18}{30} frac{3}{5}]

[cos A frac{3}{5} implies A cos^{-1}left(frac{3}{5}right)]

Step 2: Apply the cosine rule to find angle C

[cos C frac{a^2 b^2 - c^2}{2ab}]

[cos C frac{4^2 5^2 - 3^2}{2 times 4 times 5}] [cos C frac{16 25 - 9}{40} frac{32}{40} frac{4}{5}]

[cos C frac{4}{5} implies C cos^{-1}left(frac{4}{5}right)]

Thus, using both sine and cosine rules, we find the measures of angles A, B, and C.

Conclusion

In a right-angled triangle, knowing either 2 sides or one side and one additional angle is sufficient to determine the remaining sides and angles. The sine and cosine rules are powerful tools for solving such problems.

Understanding the relationships between sides and angles in right-angled triangles is fundamental in trigonometry and has wide applications in geometry, physics, and engineering.

Keywords: right-angle triangle, sine rule, cosine rule