Resolving the Angles of an Acute Triangle with Trigonometric Conditions

In this article, we will demonstrate how to resolve the angles of an acute triangle given specific conditions involving tan and sec. Specifically, we will solve the problem of finding the angles A, B, and C in an acute triangle ABC where tan(AB-C) 1 and sec(BC-A) 2.

Understanding the Problem

We are given the conditions:

In an acute triangle, all angles must be less than 90 degrees. Additionally, the sum of the angles in any triangle is always 180°.

Solving the Problem Step-by-Step

Step 1: Start with the given equation for AB-C.

tan(AB-C) 1

Since tan(45°) 1, we can conclude that:

AB-C 45°

Step 2: Express A and B in terms of C.

A B - C 45°

Step 3: Use the secant condition for BC-A.

sec(BC-A) 2

Since sec(60°) 2, we can conclude that:

BC-A 60°

Step 4: Express B and C in terms of A.

B - C A 60°

Step 5: Use the triangle sum property.

B C A 180°

Express C in terms of A and B.

C 180° - A - B

Step 6: Substitute C from Step 5 into the equation from Step 1.

A B - (180° - A - B) 45°

Simplify the equation:

2A 2B - 180° 45°

2A 2B 225°

A B 112.5°

Step 7: Substitute C from Step 5 into the equation from Step 3.

B (180° - A - B) - A 60°

180° - 2A 60°

120° 2A

A 60°

Step 8: Find B and C using A 60°.

A B 112.5°

60° B 112.5°

B 52.5°

C 180° - 60° - 52.5°

C 67.5°

Final Values:

A 60° B 52.5° C 67.5°

Conclusion

In this article, we demonstrated how to solve for the angles of an acute triangle given specific trigonometric conditions. By applying the properties of tan and sec and using logical deductions, we were able to determine the individual angles of the triangle. The key steps involve expressing the angles in terms of each other, simplifying accordingly, and using the triangle sum property to ultimately find the values.

Keywords: acute triangle, trigonometric conditions, problem solving