Solving Structural Problems Using Trigonometric Functions and Substitution

Solving Structural Problems Using Trigonometric Functions and Substitution

Understanding Trigonometry in Structual Problems

  • Trigonometry is a branch of mathematics that studies relationships involving lengths and angles of triangles.
  • In structural problems, we often deal with right-angled triangles, making trigonometry an essential tool.
  • Three main trigonometric functions are sine (sin), cosine (cos) and tangent (tan).
  • Familiarity with SOH-CAH-TOA will enable a quick recall of these relationships.

Trigonometric Functions

  • Sin of an angle in a right-angle triangle is the ratio of the length of the opposite side to the length of the hypotenuse.
  • Cos of an angle is the ratio of the length of the adjacent side to the hypotenuse.
  • Tan of an angle is the ratio of the length of the opposite side to the adjacent side.
  • Practicing the usage of these functions will aid in understanding and solving complex problems.

Applying Trigonometry in Engineering Calculations

  • You can often resolve forces in structural problems into vertical and horizontal components using trigonometric functions.
  • Understanding the vector summation of forces and their components will help balance structural systems.
  • Calculating angles and knowing their trigonometric function values may be necessary when analysing force directions.

Substituting Values

  • Following the determination of a relationship, applying substitution allows for unknown quantities to be calculated.
  • It’s vital to remember that all substituted values must share consistent units. If not, convert units before performing the substitution.
  • Keep clear track of which substitution corresponds to which variable. Muddling these up can lead to incorrect results.

Solving Problems using Trigonometric functions and Substitution

  • Start by visualising or drawing a diagram of the problem, marking all known and unknown components.
  • Identify the trigonometric relationship that might help resolve the problem.
  • Rearrange the equation, isolate the unknown element, and substitute known values.
  • Remember to handle units with caution as trigonometric functions generally deal with angles (degrees or radians) and lengths.

Checking Your Calculations

  • Cross-check all calculations for any potential errors by substituting your answers back into the original equations.
  • Double-check all trigonometric functions, their relations to the sides of triangles, and their numerical values.
  • Review if your final results make practical sense. For instance, an obtuse angle in a right-angle triangle would signify an error.