Cofunction Identities in Trigonometry (With Proof and Examples)

Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.

Cofunction identities are derived to obtain the sum and difference identities for the sine and tangent functions. Cofunction identities are derived directly from the difference identity for cosine. The cofunction identities show the relationship between sine, cosine, tangent, cotangent, secant, and cosecant. The value of an angle's trig function equals the value of the angle's complement's cofunction.

We refer to the sine and cosine functions as cofunctions of each other. Similarly, the tangent and cotangent functions are cofunctions, as are the secant and cosecant.

If variable u is the radian measure of an acute angle, then the angle measure with radian measure π/2 – u is complementary to u. We may consider the right triangle shown in the figure below.

sin (u) = a/c = cos (π/2 – u)

cos (u) = b/c = sin (π/2 – u)

tan (u) = a/b = cot (π/2 – u)

The three formulas and their analogues secant(u), cosecant(u), and cotangent(u) state that the function value of variable u is equal to the cofunction of the complementary angle π/2 – u.

Given variable u as a real number or the radian measure of an angle, then the cofunction formulas in radians are shown:

Sine and cosine are cofunctions and complements.

cos (π/2 – u) = sin (u)

sin (π/2 – u) = cos (u)

Tangent and cotangent are cofunctions and complements.

tan (π/2 – u) = cot (u)

cot (π/2 – u) = tan (u)

Secant and cosecant are cofunctions and complements.

sec (π/2−u) = csc (u)

csc (π/2−u) = sec (u)

Given variable u as a real number or the radian measure of an angle, then the cofunction formulas in degrees are shown:

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Sine and cosine are cofunctions and complements.

cos (90° – u) = sin (u)

sin (90° – u) = cos (u)

Tangent and cotangent are cofunctions and complements.

tan (90° – u) = cot (u)

cot (90° – u) = tan (u)

Secant and cosecant are cofunctions and complements.

sec (90° - u) = csc (u)

csc (90° − u) = sec (u)

Cofunction Identities Proof

Let's take a look at some proofs.

Proof 1: Cosine to Sine

Step 1: In deriving the first cofunction identity, we use the difference formula or the subtraction formula for cosine; we have

cos (π/2 – u) = cos (π/2) cos (u) + sin (π/2) sin (u)

Step 2: Evaluate the trigonometric functions that are solvable.

cos (π/2 – u) = (0) cos (u) + (1) sin (u)

Step 3: Simplify the expression. As a result, this gives us formula (1)

cos (π/2 – u) = sin (u)

Proof 2: Sine to Cosine

Step 1: We can use the result in proof 1 to prove the second cofunction identity. If we substitute π/2 – v in the first formula, we obtain

cos [π/2 – (π/2 – v)] = sin (π/2 – v)

Step 2: Evaluate the value trigonometric functions that are solvable.

cos (v) = sin (π/2 – v)

Step 3: Since the symbol v is arbitrary, the derived equation is equivalent to the second cofunction formula.

cos (u) = sin (π/2 – u)

Proof 3: Tangent to Cotangent

Step 1: Using the tangent identity, cofunction formulas 1 and 2, and the cotangent identity, we obtain proof for the third formula:

tan (π/2 – u) = [sin (π/2 – u)] / [cos (π/2 – u)]

Step 2: Simplify the trigonometric expression.

tan (π/2 – u) = cos (u) / sin (u)

tan (π/2 – u) = cot (u)

To further understand, below are some trigonometric cofunction identities examples you can run through.

Example 1: Cofunction of Sine Expressions

Find an angle θ that makes the trigonometric expression sin (θ) = cos (3θ -10) right.

Solution

Since we want cofunction values to be equal, the two angles must be complementary.

θ + (3θ - 10°) = 90°

4θ - 10° = 90°

θ = 25°

Answer

The angles θ that makes the expression true is θ = 25°.

Example 2: Cofunction of Tangent Functions

Find an angle θ that makes the trigonometric expression tan θ = cot (θ/2 + π/12) true.

Solution

Again, the two angles must be complementary. Hence,

θ + (θ/2 + π/12) = π/2

3θ/2 = π/2 – π/12 = 5π/12

3θ/2 = 5π/12

θ = 10π/36 = 5π/18

Answer

The final value of θ = 5π/18.

Example 3: Finding the Value of Angle Measure U

If cos (π/2 – u) = sin (π/8), find the value of variable u given that it lies between 0 and π/2.

Solution

Recall the cofunction identity for cosine and use it to assess the given trigonometric expressions.

cos (π/2 – u) = sin (u)

cos (π/2 – u) = sin (π/8)

u = π/8

Answer

Therefore, the value of the variable u is π/8.

Example 4: Evaluating a Function Using the Cofunction Identities

Evaluate the cosecant function cosecant (5π / 6).

Solution

Simplify the given cosecant function by transforming it to an equation with its basic equivalent which is sine.

csc (5π / 6) = 1 / sin (5π / 6)

Apply the cofunction identity for sine.

csc (5π / 6) = 1 / sin (π / 2 + π / 3)

Further simplify the expression and solve for the function.

csc (5π / 6) = 1 / sin (π / 2 – (-π / 3))

csc (5π / 6) = 1 / cos (-π / 3)

csc (5π / 6) = 1 / cos (π / 3)

csc (5π / 6) = 2

Answer

The value of csc (5π / 6) is 2.

Example 5: Finding the Value of a Tangent Function

If tan (π / 2 – x) + cot (π / 2 – x) = 2, what is the value of tan (x)?

Solution

From the trigonometric co-function identities, we know that tan (π / 2 – x) = cot (x) and cot (π / 2 – x) = tan (x). Hence, substituting to the given equation results to the following.

tan (π / 2 – u) + cot (π / 2 – x) = 2

cot (x) + tan (x) = 2

[1 / tan(x)] + tan (x) = 2

1 + tan² (x) = 2 tan (x)

tan² (x) – 2 tan (x) + 1 = 0

(tan x – 1)² = 0

tan (x) = 1

Answer

The value of tan(x) is equal to 1.

Example 6: Cofunction Identity for Secant Function

If sec (π / 2 – x) = csc (π / 8), what is the value of x, given it lies between 0 and π / 2?

Solution

sec (π / 2 – x) = csc (x)

sec (π / 2 – x) = csc (π / 8)

x = π / 8

Answer

The value of x is π / 8.

Example 7: Finding the Value of a Cotangent Function

Find the value of cot (45°).

Solution

Use the cofunction identity tan (90° - u) = cot (u) to rewrite the problem.

cot (45°) = tan (90° - 45°)

cot (45°) = tan (45°)

cot (45°) = 1

Answer

The value of cot (45°) is 1.

Example 8: Rewriting Trigonometric Equations Using Cofunction Identities

Use cofunction identities to help you write the following expressions as the function of an acute angle of measure less than 45°.

a. tan (60°)

b. sin (122°)

c. cos (285°)

d. cot (80°)

Solution

Use the tangent cofunction identity for tan (60°).

tan (u) = cot (90° - u)

tan (60°) = cot (90° – 60)

tan (60°) = cot (30°)

Apply the sine cofunction identity for sin (122°).

sin (u) = cos (90° - u)

sin (122°) = cos (90° - 122)

sin (122°) = cos (-32)

cos (-32°) = cos (32°)

Use the cosine cofunction identity for cosine.

cos (u) = sin (90° - u)

cos (285°) = sin (90° - u)

cos (285°) = sin (90° - 285°)

cos (285°) = sin (-195°)

sin (-195°) = sin (15)

Apply the cotangent cofunction identity for the last question.

cot (u) = tan (90° - u)

cot (80°) = tan (90° - 80°)

cot (80°) = tan (10°)

Answer

a. cot (30°)

b. cos (32°)

c. sin (15°)

d. tan (10°)

Example 9: Finding the Value of θ Using Cofunction Identities

Find the value of θ for which the following trigonometric expressions are true.

a. sin (θ) = cos (20°)

b. cos (θ) = sin (33°)

c. tan (θ) = cot (78°)

d. csc (θ) = sin (14°)

Solution

Use the sine cofunction identity to solve for θ.

sin (θ) = cos (20°)

cos (u) = sin (90° - u)

cos (20°) = sin (90° - 20°)

cos (20°) = sin (70°)

θ = 70°

Apply the cofunction identity for cosine in solving for θ.

cos (θ) = sin (33°)

sin (33°) = cos (90° - u)

sin (33°) = cos (90° - 33°)

sin (33°) = cos (57°)

θ = 57°

Utilize the cofunction identity for tangent in finding the value of θ.

tan (θ) = cot (78°)

cot (78°) = tan (90° - u)

cot (78°) = tan (90° - 78°)

cot (78°) = tan (12°)

θ = 12°

Use the cofunction identity for cosecant in evaluating the given the expression.

csc (θ) = sec (14°)

sec (14°) = csc (90° - u)

sec (14°) = csc (90° - 14°)

sec (14°) = csc (76°)

θ = 76°

Answer

a. θ = 70°

b. θ = 57°

c. θ = 12°

d. θ = 76°

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© 2021 Ray