Cofunction Identities in Trigonometry (With Proof and Examples)
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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:
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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