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Complement of an Angle Identities

Complement Identities (Cofunction Identities)

FunctionFormulaRadian FormDescription
sin(90° - θ)cos⁡θ\cos\thetasin(π/2 - θ) = cos θSine of complement equals cosine - fundamental cofunction relationship
cos(90° - θ)sin⁡θ\sin\thetacos(π/2 - θ) = sin θCosine of complement equals sine - shows function duality
tan(90° - θ)cot⁡θ\cot\thetatan(π/2 - θ) = cot θTangent of complement equals cotangent - reciprocal relationship
csc(90° - θ)sec⁡θ\sec\thetacsc(π/2 - θ) = sec θCosecant of complement equals secant - follows from sine-cosine relationship
sec(90° - θ)csc⁡θ\csc\thetasec(π/2 - θ) = csc θSecant of complement equals cosecant - follows from cosine-sine relationship
cot(90° - θ)tan⁡θ\tan\thetacot(π/2 - θ) = tan θCotangent of complement equals tangent - completes the cofunction pairs








The concept of Complementary Angles






The concept of Complementary Angles

In geometry, the terms complementary angles and complement of an angle are often used interchangeably, which can cause confusion. Strictly speaking, they refer to related but slightly different ideas, so it’s useful to make the distinction clear.

Complementary angles: a pair of two angles that add up to 90∘.
Example:
30° and 60° are complementary angles.

Complement of an angle: the specific other angle that makes the sum 90°.Or, in another words, a complementary angle is what you add to original one to make a right angle.
Example:
The complement of 30° is 60°.

The term complement in its strict mathematical sense only applies to acute angles (angles less than 90°).
For an angle to have a complement, both the angle and its complement must be positive and their sum must equal 90°.
So:
If θ is acute (0° < θ < 90°), then its complement (90° - θ) is also acute and positive
If θ ≥ 90°, then (90° - θ) ≤ 0°, which isn't considered a valid angle complement

This is why complement angles are typically discussed in the context of right triangles, where all angles are acute (except the right angle itself, which doesn't have a complement in the traditional sense).
x y θ 90°-θ r = 1 0° (0) (1, 0) 90° (π/2) (0, 1) 180° (π) (-1, 0) 270° (3π/2) (0, -1) 0.5 1 -0.5 -1 0.5 1 -0.5 -1
Complementary Angles in the Unit Circle

This diagram shows an acute angle θ (theta) and its complement (90° - θ) in the unit circle. The blue arc represents the angle θ measured from the positive x-axis, while the green arc shows its complement, which is the remaining angle needed to reach 90°.

Key Points:
The two angles are complementary because they add up to 90°: θ + (90° - θ) = 90°
For an angle to have a complement, it must be acute (between 0° and 90°)
The red vector has length r = 1 (unit radius)
Together, these complementary angles form a right angle (90°)

Why Acute Only?
An angle can only have a complement if both the angle and its complement are positive. If θ ≥ 90°, then (90° - θ) ≤ 0°, which is not a valid positive angle. This is why complements are only defined for acute angles.

This relationship is fundamental in trigonometry, especially when working with right triangles where the two acute angles are always complementary.