Understanding Trigonometric Functions
Trigonometric functions relate the angles of a triangle to the lengths of its sides. They are fundamental in mathematics, physics, engineering, and many other fields. The six basic trig functions are sine, cosine, tangent, cosecant, secant, and cotangent.
The Six Trig Functions
Sine (sin)
Ratio of opposite side to hypotenuse in a right triangle.
sin(x) = opposite / hypotenuse
Cosine (cos)
Ratio of adjacent side to hypotenuse in a right triangle.
cos(x) = adjacent / hypotenuse
Tangent (tan)
Ratio of opposite side to adjacent side; equals sin/cos.
tan(x) = opposite / adjacent
Cosecant (csc)
Reciprocal of sine.
csc(x) = 1 / sin(x)
Secant (sec)
Reciprocal of cosine.
sec(x) = 1 / cos(x)
Cotangent (cot)
Reciprocal of tangent; equals cos/sin.
cot(x) = 1 / tan(x)
Inverse Trigonometric Functions
Inverse trig functions (arcsin, arccos, arctan, etc.) return the angle given the ratio. For example, if sin(x) = 0.5, then arcsin(0.5) = 30 degrees. These are essential for solving equations and finding angles from known side ratios.
Common Angle Values
- sin(0°) = 0, sin(30°) = 0.5, sin(45°) = sqrt(2)/2, sin(60°) = sqrt(3)/2, sin(90°) = 1
- cos(0°) = 1, cos(30°) = sqrt(3)/2, cos(45°) = sqrt(2)/2, cos(60°) = 0.5, cos(90°) = 0
- tan(0°) = 0, tan(30°) = sqrt(3)/3, tan(45°) = 1, tan(60°) = sqrt(3)