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
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Angle Information
30.00°
0.5236 rad
I
30.00°
✨ This is a special angle with exact trigonometric values!
Trigonometric Values
Basic Functions
Reciprocal Functions
Quadrant I Sign Analysis
sin
cos
tan
Special Angles Quick Access
Unit Circle Quadrants
Quadrant I (0° - 90°)
All functions positive
sin(+), cos(+), tan(+)
Quadrant II (90° - 180°)
Only sine positive
sin(+), cos(-), tan(-)
Quadrant III (180° - 270°)
Only tangent positive
sin(-), cos(-), tan(+)
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