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.

What are Cofunction Identities?

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 Formula

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.

Cofunction Identities Proof

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)

Verification of Cofunction Identities

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.

Using Cofunction Identities

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.

ASTC rule

The concept of cofunction formulas is based on the trigonometric quadrant table. It follows the ASTC rule, which stands for the “all sin cos tan” rule. It tells which trigonometric ratios are positive in any given quadrant. It further explains that all trigonometric ratios are positive in the first quadrant, only sine and cosecant ratios are positive in the second quadrant, tangent and cotangent are positive in the third quadrant, while cosecant and secant are positive in the fourth quadrant. The first quadrant has angles from 0° to 90°; the second quadrant has 90° to 180°, the third quadrant has 180° to 270°, and the fourth quadrant has 270° to 360°.

Cofunction Identities Examples

Example 1: Determine the value of sin 150° using cofunction identities.

Solution: To find the value of sin 150°, we will use the formula sin θ = cos (90° – θ). So, we havesin 150° = cos (90° – 150°)= cos (-60°)= cos (60°) — [Because cos (-x) = cos x for all x.]= 1/2 — [Because cos 60° = 1/2]

Answer: sin 150° = 1/2

Example 2: Find the value of tan 30° + cot 150° using cofunction identities.

Solution: To find the value tan 30° + cot 150°, we will use first the values of tan 30° and cot 150°, separately.tan 30° = 1/√3cot 150° = 1 / tan 150° — [Because tan and cot are reciprocals of each other.]= 1 / tan (90° + 60°)= 1 / tan (90° – (-60°))= 1 / cot (-60°) — [Using cofunction identity cot θ = tan (90° – θ)]= – 1 / cot 60°= -1 / √3So, we have tan 300° + cot 150° = 1/√3 – 1/√3 = 0.

Answer: tan 300° + cot 150° = 0

Example 3: Find the value of θ if tan θ = cot (θ/2 + π/12) using cofunction identities.

Solution: To find the value of θ, we will use the formula tan θ = cot (π/2 – θ). So, we havetan θ = cot (θ/2 + π/12)⇒ cot (π/2 – θ) = cot (θ/2 + π/12)⇒ π/2 – θ = θ/2 + π/12⇒ θ + θ/2 = π/2 – π/12⇒ 3θ/2 = 6π/12 – π/2⇒ θ = 5π/12 × 2/3= 5π/18

Answer: θ = 5π/18

Problem 1: Calculate the value of sin 25° cos 75° + sin 75° cos 25°.

Solution:

We know,

sin 25° = cos (90° – 25°) = cos 75°

cos 25° = sin (90° – 25°) = sin 75°

So, the given expression becomes,

sin 25° cos 75° + sin 75° cos 25° = cos 75° cos 75° + sin 75° sin 75°

= cos² 75° + sin² 75°

= 1

Problem 2: Calculate the value of sin 35° cos 65° + sin 65° cos 35°.

Solution:

We know,

sin 35° = cos (90° – 35°) = cos 65°

cos 35° = sin (90° – 35°) = sin 65°

So, the given expression becomes,

sin 35° cos 65° + sin 65° cos 35° = cos 65° cos 65° + sin 65° sin 65°

= cos² 65° + sin² 65°

= 1

Problem 3: Calculate the value of sec 20° cosec 70° – tan 20° cot 70°.

Solution:

cosec 70° = sec (90° – 70°) = sec 20°

cot 70° = tan (90° – 70°) = tan 20°

So, the given expression becomes,

sec 20° cosec 70° – tan 20° cot 70° = sec 20° sec 20° – tan 20° tan 20°

= sec² 20° – tan² 20°

= 1

Problem 4: Calculate the value of cosec 40° sec 50° – cot 40° tan 50°.

Solution:

sec 50° = cosec (90° – 50°) = cosec 40°

tan 50° = cot (90° – 50°) = cot 40°

So, the given expression becomes,

cosec 40° sec 50° – cot 40° tan 50° = cosec 40° cosec 40° – cot 40° cot 40°

= cosec² 40° – cot² 40°

= 1

Problem 5: Calculate the value of tan 1° tan 2° tan 3°… tan 89°.

Solution:

We have,

A = tan 1° tan 2° tan 3°……. tan 89°

= tan 1° tan 2° tan 3°……. tan 87° tan 88° tan 89°

= tan 1° tan 2° tan 3°….. tan 45° ….. tan 87° tan 88° tan 89°

= tan 1° tan 2° tan 3°….. tan 45° ….. cot 3° cot 2° cot 1°

= tan 1° tan 2° tan 3°….. tan 45° ….. (1/tan 3°) (1/tan 2°) (1/tan 1°)

= tan 45°

= 1

Problem 6: Calculate the value of cot 23° cot 41° cot 60° cot 67° cot 49°.

Solution:

cot 23° = tan (90 – 23) = tan 67°

cot 41° = tan (90 – 41) = tan 49°

So, the given expression becomes,

cot 23° cot 41° cot 60° cot 67° cot 49° = tan 67° tan 49° cot 60° cot 67° cot 49°

= tan 67° tan 49° cot 60° (1/tan 67°) (1/tan 49°)

= cot 60°

= 1/√3

Problem 7: Calculate the value of x for the equation, sin 2x = cos (x – 30°).

Solution:

We have,

sin 2x = cos (x – 30°)

Using the identity cos x = sin (90° – x) we get,

sin 2x = sin (90° – (x – 30°))

sin 2x = sin (90° – x + 30°)

sin 2x = sin (120° – x)

=> 2x = 120° – x

=> 3x = 120°

=> x = 40°

FAQs on Cofunction Identities

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.

What are Cofunction Formulas in Trigonometry?

Cofunction Formulas are trigonometric identities that establish relationships between the trigonometric functions of complementary angles.

How do Cofunction Identities work?

Cofunction Identities relate the trigonometric functions of complementary angles, simplifying calculations and problem-solving.

How do Cofunction Identities work?

Cofunction Identities relate the trigonometric functions of complementary angles, simplifying calculations and problem-solving.

How do Cofunction Formulas simplify trigonometric equations?

Cofunction Formulas allow us to transform trigonometric expressions involving one function into those of its complementary function, making calculations more straightforward.

What is the significance of periodicity in Cofunction Formulas?

Periodicity, the repeating pattern of trigonometric functions, is preserved in Cofunction Formulas, aiding in graphing and analysing functions over multiple cycles.

What is the meaning of trigonometric cofunction formulas?

Trigonometric cofunction formulas are identities that relate the trigonometric functions of complementary angles, simplifying calculations and problem-solving.

How can Cofunction Formulas be used in trigonometric proofs?

Cofunction Formulas are instrumental in proving various trigonometric identities and equations through algebraic manipulations and substitutions.