Trigonometry

Basic Trigonometry Functions

• The three basic trigonometric functions are sinus (sin), cosine (cos), and tangent (tan).
• In a right angled triangle, sin of an angle θ is the ratio of the length of the side opposite to the angle θ to the length of the hypotenuse. cos of an angle θ is the ratio of the length of the adjacent side to the angle θ to the length of the hypotenuse. tan of an angle θ is the ratio of the sin(θ) to the cos(θ).
• Each of these functions forms a pattern that is repeated every 360 degrees (or 2π radians).

Reciprocal Trigonometric Functions

• The reciprocal of the sinus function is the cosecant (csc), the reciprocal of the cosine function is the secant (sec) and the reciprocal of the tangent function is the cotangent (cot).
• These reciprocal functions are less commonly used, but can still be useful in certain trigonometric equations or expressions.

Pythagorean Identities

• From the definition of sin and cos in a right triangle, we derive the Pythagorean identity: sin²(θ) + cos²(θ) = 1.
• Two other identities can be derived from this: 1 + tan²(θ) = sec²(θ) and 1 + cot²(θ) = csc²(θ). These identities are important for rearranging and simplifying trigonometric expressions.

Trigonometric Function Graphs

• The graphs of trigonometric functions are periodic, repeating every 2π radians (or 360 degrees).
• The graph of y = sin x is a wave that passes through the origin, has a maximum value of 1 and a minimum value of -1.
• The graph of y = cos x has the same shape, but begins at its maximum point when x = 0.
• The graph of y = tan x has asymptotes where the function is undefined (at odd multiples of π/2).

Trigonometric Equations and Inequalities

• Trigonometric equations are solved by using algebraic techniques and trigonometric identities.
• Common methods include factoring, using fundamental identities, and using double or half angle identities.
• Inequalities can be solved by finding the critical values of the inequality and testing intervals.

Compound Angle Formulas

• The sine of sum of two angles can be given as: sin(A + B) = sin A cos B + cos A sin B
• The cosine of sum of two angles is: cos(A + B) = cos A cos B - sin A sin B
• These identities are useful when dealing with complex trigonometric expressions or when simplifying calculations.

Applications of Trigonometry

• Trigonometry has many applications in various fields such as physics, engineering, astronomy, music and even medicine.
• For instance, it can be used to calculate distances and angles in navigation and in constructing buildings, or to predict the behavior of waves such as light or sound waves.
• In the field of statistics and probability, the trigonometric functions form the basis for the Normal Distribution curve, used in theories of probability and statistics.