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.

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.

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

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.

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.

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.

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.