In geometry and trigonometry, “specific angles” (most commonly referred to as “special angles”) are angles that have exact, easily calculated trigonometric values. Instead of using long decimals, mathematicians use these exact numbers—often involving fractions and square roots (radicals)—to solve equations quickly.
The primary special angles found on the unit circle are 0°, 30°, 45°, 60°, and 90°. The 5 Primary Special Angles
The exact values for the three main trigonometric functions—Sine ( ), Cosine ( ), and Tangent ( tantangent )—for these specific angles are outlined below: Angle in Degrees (°) Angle in Radians (rad) tantangent 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
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 Why Are These Angles “Special”?
These specific measurements are derived from two geometric shapes that allow us to use Pythagoras’ Theorem to find exact side lengths:
The 45°-45°-90° Triangle: Created by cutting a square in half diagonally. The two legs are equal in length, and the hypotenuse is always 2the square root of 2 end-root times the length of a leg.
The 30°-60°-90° Triangle: Created by cutting an equilateral triangle exactly down the middle. The shortest side is always half the length of the hypotenuse, and the middle side is 3the square root of 3 end-root times the shortest side. Extension to Other Quadrants
These rules also apply to larger specific angles across a full 360° circle (like 120°, 135°, 150°, 180°, 270°, and 360°). They share the exact same reference values as the primary five, but their positive or negative signs change depending on which quadrant they fall into on a graph.
If you want to solve a specific problem or see how these values are calculated: Provide the angle measurement you are looking at.
Specify the trigonometric function (e.g., Sine, Cosine, Tangent) you need to apply.I will walk you through the exact step-by-step solution. Unit 1: Special angles
There are three special angles. They are 30∘ , 45∘ and 60∘ . They are special because they provide complementary pairs of angles (
Special Angles — Trig Values, Table & Examples – Mathwords