Exact Value of Trig Functions Calculator

Find exact trigonometric values for special angles with step-by-step explanations

Calculate Exact Trigonometric Values

Enter angle in degrees or radians

Choose which functions to display

Angle Information

Degrees:

30.00°

Radians:

0.5236 rad

Quadrant:

I

Reference Angle:

30.00°

✨ This is a special angle with exact trigonometric values!

Trigonometric Values

Basic Functions

1/2
sin(30°)
0.5000
√3/2
cos(30°)
0.8660
√3/3
tan(30°)
0.5774

Reciprocal Functions

2.000000
csc(30°)
= 1/sin(30°)
1.154701
sec(30°)
= 1/cos(30°)
1.732051
cot(30°)
= 1/tan(30°)

Quadrant I Sign Analysis

+

sin

+

cos

+

tan

Special Angles Quick Access

Unit Circle Quadrants

I

Quadrant I (0° - 90°)

All functions positive

sin(+), cos(+), tan(+)

II

Quadrant II (90° - 180°)

Only sine positive

sin(+), cos(-), tan(-)

III

Quadrant III (180° - 270°)

Only tangent positive

sin(-), cos(-), tan(+)

IV

Quadrant IV (270° - 360°)

Only cosine positive

sin(-), cos(+), tan(-)

Special Right Triangles

30°-60°-90° Triangle

Sides ratio: 1 : √3 : 2

sin(30°) = 1/2, cos(30°) = √3/2

sin(60°) = √3/2, cos(60°) = 1/2

45°-45°-90° Triangle

Sides ratio: 1 : 1 : √2

sin(45°) = √2/2

cos(45°) = √2/2

tan(45°) = 1

Understanding Exact Trigonometric Values

What are Exact Values?

Exact trigonometric values are precise mathematical expressions (often involving square roots) for specific angles, rather than decimal approximations. These values are derived from special right triangles and the unit circle.

Common Special Angles

  • 0°, 90°, 180°, 270°: Quadrant boundary angles
  • 30°, 60°: From 30-60-90 triangle
  • 45°: From 45-45-90 triangle
  • 15°, 75°: From angle addition formulas

Key Formulas

Pythagorean Identity:
sin²(θ) + cos²(θ) = 1

Half-Angle Formulas:
sin(θ/2) = ±√[(1-cos(θ))/2]
cos(θ/2) = ±√[(1+cos(θ))/2]

Double-Angle Formulas:
sin(2θ) = 2sin(θ)cos(θ)
cos(2θ) = cos²(θ) - sin²(θ)

Reference Angles

The reference angle is the acute angle between the terminal side of an angle and the x-axis. It helps determine the exact values in different quadrants by applying appropriate signs.

Memory Aid: "All Students Take Calculus" - All positive in Q1, Sine in Q2, Tangent in Q3, Cosine in Q4