# 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.