Exact Values of Trigonometric Functions

Find exact values of sin, cos, tan, csc, sec, and cot at standard angles with radical expressions.

Select Mode & Enter Angle

Quick select standard angles:

Result

Angle: 45 degrees (pi/4 radians)
sin(45) = sqrt(2)/2
exact value
Angle (degrees)45
Angle (radians)pi/4
QuadrantI
Reference Angle45

Step-by-Step Solution

sin(45) = sqrt(2)/2 = 0.707107

Understanding Exact Trigonometric Values

Certain angles produce trigonometric values that can be expressed exactly using radicals (square roots) and rational numbers rather than decimal approximations. These "special angles" include 0, 30, 45, 60, and 90 degrees and their multiples. Knowing these exact values is fundamental in mathematics, physics, and engineering.

The exact values come from the properties of special right triangles: the 30-60-90 triangle (with sides 1, sqrt(3), 2) and the 45-45-90 triangle (with sides 1, 1, sqrt(2)).

Standard Angle Exact Values

0 degrees (0 rad)

sin=0, cos=1, tan=0

sin(0)=0, cos(0)=1, tan(0)=0

30 degrees (pi/6)

From the 30-60-90 triangle.

sin=1/2, cos=sqrt(3)/2, tan=sqrt(3)/3

45 degrees (pi/4)

From the isosceles right triangle.

sin=sqrt(2)/2, cos=sqrt(2)/2, tan=1

60 degrees (pi/3)

From the 30-60-90 triangle.

sin=sqrt(3)/2, cos=1/2, tan=sqrt(3)

90 degrees (pi/2)

Tangent is undefined at this angle.

sin=1, cos=0, tan=undefined

Unit Circle

All trig values can be read from the unit circle coordinates (cos, sin).

(cos(t), sin(t)) on x^2+y^2=1

Quadrant Sign Rules (ASTC)

The signs of trigonometric functions depend on the quadrant. Remember "All Students Take Calculus" (ASTC): All functions are positive in Quadrant I, only Sine in QII, only Tangent in QIII, and only Cosine in QIV.

Tips for Exact Values

  • Memorize the values for 0, 30, 45, 60, and 90 degrees; derive the rest using reference angles and quadrant signs.
  • The reference angle is always the acute angle formed with the x-axis.
  • Reciprocal functions: csc = 1/sin, sec = 1/cos, cot = 1/tan.
  • Radians and degrees: pi radians = 180 degrees.
  • Co-function identities: sin(90-x) = cos(x), tan(90-x) = cot(x).