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Pre-Calculus Trigonometric identities

Cofunction and even/odd identities

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Cofunction and Even/Odd Identities

Common Core Pre-Calculus • Standard F-TF.4 • Trigonometric Identities

Cofunction and Even/Odd Identities is a topic in Trigonometric Identities in the Common Core State Standards. It is aligned to Standard F-TF.4, which requires students to use the periodicity and symmetry of the trigonometric functions.

Cofunction identities relate a function to the cofunction of its complement \((\sin\theta=\cos(90^\circ-\theta))\), and even/odd identities describe the effect of a negative angle.

Common Core Pre-Calculus › Trigonometric Identities › Cofunction and Even/Odd Identities  —  Standard F-TF.4

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Theory

Two symmetry families simplify trig expressions:

  • Cofunction identities link a function of \(\theta\) to the cofunction of its complement \(90^\circ-\theta\): \(\sin\theta=\cos(90^\circ-\theta)\), \(\tan\theta=\cot(90^\circ-\theta)\), and so on.
  • Even/odd identities describe symmetry in the sign of the angle: \(\cos\) is even (\(\cos(-\theta)=\cos\theta\)); \(\sin\) and \(\tan\) are odd (\(\sin(-\theta)=-\sin\theta\), \(\tan(-\theta)=-\tan\theta\)).
“Co” means complement. Sine and cosine, tangent and cotangent, secant and cosecant are cofunction pairs summing to \(90^\circ\).
Special right triangle In a right triangle the two acute angles are complementary, so a function of one equals the cofunction of the other. adjacentoppositeθ
The two acute angles of a right triangle are complementary, so each function equals its cofunction of the other angle.
Even and odd symmetry on the unit circle Reflecting an angle across the x-axis keeps cosine the same but negates sine, so cosine is even and sine is odd. θ−θ
Reflecting \(\theta\) to \(-\theta\) keeps \(\cos\) but negates \(\sin\): cosine is even, sine is odd.

Cofunction and even/odd identities:

\[\sin\theta=\cos(90^\circ-\theta),\qquad \tan\theta=\cot(90^\circ-\theta)\]
\[\cos(-\theta)=\cos\theta,\quad \sin(-\theta)=-\sin\theta,\quad \tan(-\theta)=-\tan\theta\]
sine theta equals cosine of ninety minus theta; cosine is even, sine and tangent are odd
In radians the complement is \(\dfrac{\pi}{2}-\theta\).

How to apply these identities

  1. Cofunction: swap a function for its cofunction of the complementary angle.
  2. Even: drop a negative inside \(\cos\) (or \(\sec\)).
  3. Odd: pull a negative out of \(\sin,\tan,\csc,\cot\).
Example 1 — A cofunction relationship
Rewrite \(\sin 63^\circ\) as a cosine.
Solution

Cofunctions of complementary angles are equal: \(\sin\theta=\cos(90^\circ-\theta)\).

\(\sin 63^\circ\)\(=\)\(\cos(90^\circ-63^\circ)\)
\(=\)\(\cos 27^\circ\)
sine of 63 degrees equals cosine of 27 degrees
Example 2 — Even function
Simplify \(\cos(-\theta)\).
Solution

Cosine is even, so a sign inside does nothing.

\(\cos(-\theta)\)\(=\)\(\cos\theta\)
cosine of negative theta equals cosine theta
Example 3 — Odd function
Simplify \(\sin(-\theta)+\tan(-\theta)\).
Solution

Sine and tangent are odd, so each picks up a minus sign.

\(\sin(-\theta)+\tan(-\theta)\)\(=\)\(-\sin\theta-\tan\theta\)
\(=\)\(-(\sin\theta+\tan\theta)\)
equals negative the quantity sine theta plus tangent theta
Example 4 — Evaluate with even/odd
Find \(\cos\!\left(-\dfrac{\pi}{3}\right)\).
Solution

Cosine is even, so drop the negative and evaluate.

\(\cos\!\left(-\dfrac{\pi}{3}\right)\)\(=\)\(\cos\dfrac{\pi}{3}=\dfrac{1}{2}\)
cosine of negative pi over three is one half

Common pitfalls

Only cosine (and secant) are even. Sine, tangent, cosecant, and cotangent are odd — the sign comes out.
Cofunctions use the complement, \(90^\circ-\theta\), not the supplement \(180^\circ-\theta\).
Even/odd is about the sign of the angle, not the sign of the output.

Frequently asked questions

What is a cofunction identity?

A function of an angle equals the cofunction of its complement, e.g. \(\sin\theta=\cos(90^\circ-\theta)\).

Which trig functions are even and which are odd?

Cosine and secant are even; sine, cosecant, tangent, and cotangent are odd.

What does an odd identity let you do?

Pull the negative out: \(\sin(-\theta)=-\sin\theta\) and \(\tan(-\theta)=-\tan\theta\).

Why are these called cofunctions?

Because “co” stands for complement: the pairs (sine, cosine), (tangent, cotangent), (secant, cosecant) relate angles that add to \(90^\circ\).

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