Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles. The word comes from the Greek words "trigonon" (triangle) and "metron" (measure).
The three primary trigonometric functions - sine, cosine, and tangent - are defined based on the ratios of sides in a right triangle. These functions are fundamental to fields including physics, engineering, navigation, and signal processing.
In a right triangle with angle θ:
The reciprocal functions are:
For inverse trigonometric functions, the output is an angle whose trigonometric function equals the input value.
Problem: Find the sine of 30 degrees.
Solution:
Problem: Find the cosine of 60 degrees.
Solution:
Degrees and radians are two units for measuring angles. A full circle is 360 degrees or 2π radians. To convert: radians = degrees × (π/180), or degrees = radians × (180/π).
tan(θ) = sin(θ)/cos(θ). At 90°, cos(90°) = 0, so dividing by zero makes tan(90°) undefined. On a graph, the tangent function has a vertical asymptote at 90°.
Inverse trig functions (arcsin, arccos, arctan) find the angle when given a ratio. For example, if sin(θ) = 0.5, then θ = arcsin(0.5) = 30°.
Key values: sin(0°) = 0, sin(30°) = 0.5, sin(45°) = √2/2 ≈ 0.707, sin(60°) = √3/2 ≈ 0.866, sin(90°) = 1. The same values apply to cosine but in reverse order.