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In mathematics—specifically in geometry and trigonometry—the term “specific angles” (often referred to as “special angles”) denotes highly notable angle measurements that yield exact, easily computable trigonometric values without relying on a decimal calculator. The 5 Foundational Specific Angles

The most common specific angles used in the first quadrant of the Cartesian coordinate system include:

0° (0 radians): Represents a completely closed angle or zero rotation. 30° (

π6the fraction with numerator pi and denominator 6 end-fraction radians): Derived from a bisected equilateral triangle. 45° (

π4the fraction with numerator pi and denominator 4 end-fraction radians): Derived from an isosceles right triangle. 60° (

radians): The interior angle measure of any equilateral triangle. 90° (

π2the fraction with numerator pi and denominator 2 end-fraction radians): A right angle representing perpendicular lines. Trigonometric Values of Specific Angles

Because these angles are geometric constants, their core ratios—Sine ( ), Cosine ( ), and Tangent ( tantangent )—can be written out cleanly using exact radicals. in Degrees) in Radians) 0° 30°

π6the fraction with numerator pi and denominator 6 end-fraction 12one-half

32the fraction with numerator the square root of 3 end-root and denominator 2 end-fraction

13the fraction with numerator 1 and denominator the square root of 3 end-root end-fraction

33the fraction with numerator the square root of 3 end-root and denominator 3 end-fraction 45°

π4the fraction with numerator pi and denominator 4 end-fraction

22the fraction with numerator the square root of 2 end-root and denominator 2 end-fraction

22the fraction with numerator the square root of 2 end-root and denominator 2 end-fraction 60°

π3the fraction with numerator pi and denominator 3 end-fraction

32the fraction with numerator the square root of 3 end-root and denominator 2 end-fraction 12one-half 3the square root of 3 end-root 90°

π2the fraction with numerator pi and denominator 2 end-fraction Undefined General Classifications of Angles

When an angle is evaluated by its raw measurement rather than its precise trigonometric values, it falls into a broader descriptive category: Types of Angles (Acute, Obtuse, Right, Straight, Reflex)