Cofunction Identities

Cofunction identities in trigonometry give the relationship between the different trigonometric functions and their complementary angles. Let us recall the meaning of complementary angles. Two angles are said to be complementary angles if their sum is equal to π/2 radians or 90°. Cofunction identities are trigonometric identities that show the relationship between trigonometric ratios pairwise (sine and cosine, tangent and cotangent, secant and cosecant). We use the angle sum property of a triangle to derive the six cofunction identities.

In this article, we will derive the cofunction identities and verify them using the sum and difference formulas of trigonometric functions. We will also solve various examples to understand the usage of these cofunction identities to solve various math problems involving trigonometric functions.

Cofunction identities are trigonometric identities that show a relationship between complementary angles and trigonometric functions. We have six such identities that can be derived using a right-angled triangle, the angle sum property of a triangle, and the trigonometric ratios formulas. The cofunction identities give a relationship between trigonometric functions sine and cosine, tangent and cotangent, and secant and cosecant. These functions are referred to as cofunctions of each other. We can also derive these identities using the sum and difference formulas if trigonometric as well. Alternatively, we can use the sum and difference formulas to verify the cofunction identities.

Cofunction identities give a relationship between trigonometric functions pairwise and their complementary angles as below:

• Sine function and cosine function
• Tangent function and cotangent function
• Secant Function and Cosecant Function

Two angles are said to be complementary if their sum is 90 degrees. We can write the cofunction identities in terms of radians and degrees as these are the two units of angle measurement. The six cofunction identities are given in the table below in radians and degrees:

Let us derive these cofunction identities in the next section.

Now that we have discussed the cofunction identities in the previous section, let us now derive them using the right angle triangle. Consider a right-angled triangle ABC right angled at B. Assume angle C = θ, then using the angle sum property of a triangle we have,

∠A + ∠B + ∠C = 180°

⇒ ∠A + 90° + ∠C = 180° --- [Because angle B is a right angle]

⇒ ∠A + ∠C = 180° - 90°

⇒ ∠A + θ = 90°

⇒ ∠A = 90° - θ

Therefore, we have the three angles of the triangle ABC as ∠A = 90° - θ, ∠B = 90° and ∠C = θ. Now, let us recall the formulas of trigonometric formulas below:

• sin x = Opposite Side / Hypotenuse
• cos x = Adjacent Side / Hypotenuse
• tan x = Opposite Side / Adjacent Side
• cot x = Adjacent Side / Opposite Side
• sec x = Hypotenuse / Opposite Side
• csc x = Hypotenuse / Adjancent Side

Now, using the above formulas, we can determine the cofunction identities for triangle ABC.

• cos θ = BC / AC = sin (90° - θ)
• sin θ = AB / AC = cos (90° - θ)
• tan θ = AB / BC = cot (90° - θ)
• cot θ = BC / AB = tan (90° - θ)
• sec θ = AC / BC = csc (90° - θ)
• csc θ = AC / AB = sec (90° - θ)

Hence, we have derived the cofunction identities. To get these identities in radians, we can simply replace 90° with π/2 and get the identities as:

• cos θ = BC / AC = sin (π/2 - θ)
• sin θ = AB / AC = cos (π/2 - θ)
• tan θ = AB / BC = cot (π/2 - θ)
• cot θ = BC / AB = tan (π/2 - θ)
• sec θ = AC / BC = csc (π/2 - θ)
• csc θ = AC / AB = sec (π/2 - θ)

Now that we have proved the cofunction identities, let us verify them using the sum and difference formulas of trigonometry. We will use the following formulas to verify the identities:

• sin(A - B) = sinA cosB - cosA sinB
• cos(A - B) = cosA cosB + sinA sinB
• tan A = sin A / cos A

Expand sin (π/2 - θ), cos (π/2 - θ), and tan (π/2 - θ) using the above formulas.

• sin (π/2 - θ) = sin(π/2) cosθ - cos(π/2) sinθ
  = 1 × cos θ - 0 × sin θ --- [Because sin (π/2) = 1 and cos (π/2) = 0]
  = cos θ
• cos (π/2 - θ) = cos(π/2) cosθ + sin(π/2) sinθ
  = 0 × cos θ + 1 × sin θ --- [Because sin (π/2) = 1 and cos (π/2) = 0]
  = sin θ
• tan(π/2 - θ) = [sin (π/2 - θ)] / [cos (π/2 - θ)]
  = cos θ / sin θ
  = cot θ

Let us now verify the cofunction identities for sec, csc, and cot using reciprocal identities

• cot (π/2 - θ) = 1 / tan (π/2 - θ)
  = 1 / cot θ
  = tan θ
• sec (π/2 - θ) = 1 / cos (π/2 - θ)
  = 1 / sin θ
  = csc θ
• csc (π/2 - θ) = 1 / sin (π/2 - θ)
  = 1 / cos θ
  = sec θ

Hence, we have verified all six cofunction identities using trigonometric formulas.

Now that we have derived the formulas for the cofunction identities, let us solve a few problems to understand its application.

Example 1: Find the value of acute angle x, if sin x = cos 20°.

Solution: Using cofunction identity, cos (90° - θ) = sin θ, we can write sin x = cos 20° as

sin x = cos 20°

⇒ cos (90° - x) = cos 20°

⇒ 90° - x = 20°

⇒ x = 90° - 20°

⇒ x = 70°

Answer: Value of x is 70° if sin x = cos 20°.

Example 2: Evaluate the value of x, if sec (5x) = csc (x + 18°), where 5x is an acute angle.

Solution: To find the value of x, we will use the cofunction identity csc (90° - θ) = sec θ. We can write

sec (5x) = csc (x + 18°)

⇒ csc (90° - 5x) = csc (x + 18°)

⇒ 90° - 5x = x + 18° --- [Because it is given 5x is acute]

⇒ 5x + x = 90° - 18°

⇒ 6x = 72°

⇒ x = 72° / 6

⇒ x = 12°

Answer: Value of x is 12° if sec (5x) = csc (x + 18°), where 5x is an acute angle.

Important Notes on Cofunction Identities

• Cofunction identities show the relationship between trigonometric functions and complementary angles.
• We have main six cofunction identities:
  • cos θ = sin (90° - θ)
  • sin θ = cos (90° - θ)
  • tan θ = cot (90° - θ)
  • cot θ = tan (90° - θ)
  • sec θ = csc (90° - θ)
  • csc θ = sec (90° - θ)
• These identities can be derived using the angle sum property of a right triangle and sum and difference formulas.

☛ Related Topics:

• Sum to Product Formulas
• Hyperbolic Functions
• Derivative of Hyperbolic Functions

What are Cofunction Identities in Trigonometry?

Cofunction identities in trigonometry are formulas that show the relationship between trigonometric functions and their complementary angles pairwise - (sine and cosine, tangent and cotangent, secant and cosecant). We have mainly six cofunction identities that are used to solve various problems in trigonometry.

What are the Main Six Cofunction Identities?

The six main cofunction identities are:

• cos θ = sin (90° - θ)
• sin θ = cos (90° - θ)
• tan θ = cot (90° - θ)
• cot θ = tan (90° - θ)
• sec θ = csc (90° - θ)
• csc θ = sec (90° - θ)

We can write these identities using the measure of radians also as given below:

• cos θ = sin (π/2 - θ)
• sin θ = cos (π/2 - θ)
• tan θ = cot (π/2 - θ)
• cot θ = tan (π/2 - θ)
• sec θ = csc (π/2 - θ)
• csc θ = sec (π/2 - θ)

How Do You Find Cofunction Identities?

We can derive the formulas for the six cofunction identities using a right-angled triangle and the angle sum property of a triangle. We can also prove these identities using the sum and difference formulas and reciprocal identities in trigonometry.

What are Cofunction Identities For Tangent and Cotangent?

The cofunction identities for tangent and cotangent are given below:

• tan θ = cot (π/2 - θ)
• cot θ = tan (π/2 - θ)

We can also write these formulas in terms of degrees also as:

• tan θ = cot (90° - θ)
• cot θ = tan (90° - θ)

Why are Cofunction Identities True for all Right Triangles?

We say that two functions are cofunctions of each other if their angles are complementary, that is, the sum of their angles is π/2 rad. In an arbitrary right triangle, since one angle is π/2 rad, the sum of the other two angles is always π/2 using the nagle sum property. So, the cofunction identities are true for all right triangles and they can be easily derived using a right triangle and applying trigonometric ratios formulas to it.

When to Use Cofunction Identities?

We can use cofunction identities to simplify various complex trigonometric problems. They are used when the angles involved are complementary, that is, their sum is 90 degrees. Cofunction identities can be used to find values of trigonometric ratios with angles more than 90 degrees to simplify them.