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

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

⇒ cos (90° - x) = cos 20°

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]

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)

  ☛ Related Topics:

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

  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 have sin 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/√3 cot 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 / √3 So, 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 have tan 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

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