Proofs and Problems

Proofs and Problems

Proofs in Trigonometry

Fundamental Understanding

  • Establish a solid knowledge of key trigonometric identities like Pythagorean, reciprocal, quotient, co-function and negative angle identities.
  • Understand the definitions and properties of trigonometric functions and the relationships among them.

Proving Trigonometric Identities

  • Grasp the techniques to prove trigonometric identities. Start by noting the given identities, then use known identities, and algebraic manipulations to reach the desired results.
  • Work with one side of the equation by expanding, factoring, adding or subtracting fractions, using common denominators, or employing the Pythagorean identities to prove trigonometric identities.
  • Remember that the end goal is to manipulate one side of the equation so it matches the other side.

Proving Statements Involving Trigonometric Functions

  • Use the laws of sine and cosine to prove statements about triangles, especially the ambiguous case of the law of sines.
  • Apply the double-angle and half-angle formulas to prove statements and simplify expressions.

Solving Problems in Trigonometry

Concepts and Strategies

  • Understand how to apply trigonometric identities and formulas to solve a variety of mathematical problems.
  • Learn to solve trigonometric equations for an unknown variable using techniques such as factoring, applying identities, and squaring both sides.

Solving Trigonometric Equations

  • Grasp how to find all solutions of trigonometric equations in an interval by using unit circle, graphical interpretations, or trigonometric tables.
  • Employ the zero-product property and the quadratic formula for solving higher degree trigonometric equations.

Word Problems Involving Trigonometry

  • Gain competency in translating real world problems into trigonometric problems and solving them.
  • Use the law of sines and law of cosines to solve problems involving distances, angles, and lengths in the real world such as navigation, architecture, and physics.
  • Incorporate inverse trigonometric functions to find missing angles in application problems.
  • Learn to make use of the right triangle trigonometry in dealing with problems involving heights and distances.

Numerical Problems

  • Practice and master the use of the trigonometric ratios to find missing sides in right triangles.
  • Employ calculator applications to solve trigonometric equations numerically to a given number of decimal places.