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\nCofunction Identities Calculator
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Evaluate trig functions without a calculator using cofunction identities
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Results
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sin(θ) =
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cos(90° – θ) =
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Observation:
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Identity Used:
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\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nEvaluate trig functions without using a calculator using cofunction identities\nCofunction identities are fundamental trigonometric relationships that allow you to evaluate trigonometric functions of angles without needing a calculator. They are based on the properties of complementary angles—angles that add up to 90° (or π/2 radians). This guide will explain what cofunction identities are, how to use them, and provide practical examples to help you master trigonometric evaluations.\n\nWhat are cofunction identities?\nCofunction identities are pairs of trigonometric functions that are equal when one angle is the complement of the other. In simple terms, if you have two angles that add up to 90°, the sine of one angle is equal to the cosine of the other, the tangent of one equals the cotangent of the other, and so on.\n\nThe six trigonometric cofunction identities are:\n\nsin(θ) = cos(90° - θ)\ncos(θ) = sin(90° - θ)\ntan(θ) = cot(90° - θ)\ncot(θ) = tan(90° - θ)\nsec(θ) = csc(90° - θ)\ncsc(θ) = sec(90° - θ)\n\nWhere θ is any angle.\n\nThese identities hold true for all angles, whether they are acute (between 0° and 90°), obtuse (between 90° and 180°), or any other angle in the unit circle. They are particularly useful for simplifying trigonometric expressions and evaluating functions of angles that are not standard values.\n\nHow do cofunction identities work?\nThe basis of cofunction identities lies in the geometry of right triangles. Consider a right triangle with angles θ, 90° - θ, and 90°. The sides opposite these angles are a, b, and c (hypotenuse), respectively.\n\nThe trigonometric ratios for angle θ are:\n\nsin(θ) = opposite/hypotenuse = a/c\ncos(θ) = adjacent/hypotenuse = b/c\ntan(θ) = opposite/adjacent = a/b\n\nThe trigonometric ratios for the complementary angle 90° - θ are:\n\nsin(90° - θ) = opposite/hypotenuse = b/c\ncos(90° - θ) = adjacent/hypotenuse = a/c\ntan(90° - θ) = opposite/adjacent = b/a\n\nBy comparing these ratios,