Write The Following Function In Terms Of Its Cofunction.

Write the Following Function in Terms of Its Cofunction

Understanding how to rewrite a trigonometric function using its cofunction is a fundamental skill in pre‑calculus and calculus courses. Cofunction identities link pairs of functions whose arguments are complementary angles—angles that add up to 90° (or π⁄2 radians). Mastering these relationships not only simplifies algebraic manipulations but also prepares you for solving integrals, proving identities, and analyzing periodic phenomena. In this guide you will learn the theory behind cofunctions, see step‑by‑step procedures, work through detailed examples, and discover common pitfalls to avoid.

Introduction to Cofunctions

In trigonometry, six primary functions are defined for an angle θ: sine (sin), cosine (cos), tangent (tan), secant (sec), cosecant (csc), and cotangent (cot). Each of these has a cofunction that yields the same value when the input angle is replaced by its complement (90° − θ). The term cofunction comes from the Latin co‑ meaning “together” and function, indicating that the two functions work together via complementary angles.

The core idea can be summarized in one sentence: the value of a trigonometric function at an angle equals the value of its cofunction at the complementary angle. This principle is the basis for the identities you will use to “write the following function in terms of its cofunction.”

Cofunction Identities (The Core Formulas)

Below are the six standard cofunction identities, valid for any angle θ measured in degrees or radians (provided the functions are defined).

When working in radians, replace 90° with π⁄2:

• sin θ = cos(π⁄2 − θ)
• cos θ = sin(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)

These identities are derived from the geometry of a right triangle or the unit circle, where the acute angles of a right triangle always sum to π⁄2.

Step‑by‑Step Procedure: Writing a Function in Terms of Its Cofunction

Follow these five steps to rewrite any given trigonometric expression using its cofunction. The process works for simple functions as well as for more complex expressions involving sums, differences, or multiples of angles.

1. Identify the target function
   Determine which of the six trigonometric functions appears in the original expression (sin, cos, tan, sec, csc, or cot).

2. Locate its cofunction partner
   Use the table above to find the corresponding cofunction (e.g., if the function is sin, its cofunction is cos).

3. Replace the angle with its complement Substitute the original angle θ with (90° − θ) or (π⁄2 − θ) inside the cofunction.
   Important: Keep any coefficients, signs, or additional terms attached to θ; only the angle itself changes.

4. Simplify if necessary
   Apply algebraic rules (distributive property, factoring, etc.) to clean up the expression.
   If the original expression contained a negative sign or a coefficient, carry it through unchanged.

5. Verify the transformation
   Optionally, test the identity with a specific angle (e.g., θ = 30°) to ensure both sides produce the same numeric value.

Worked Examples

Example 1: Simple Function

Problem: Write sin θ in terms of its cofunction.

Solution:

1. Target function: sin θ.
2. Cofunction partner: cos.
3. Replace θ with its complement: cos(90° − θ).
4. No further simplification needed.

Answer: sin θ = cos(90° − θ).

Check: Let θ = 20°. sin 20° ≈ 0.342; cos(70°) ≈ 0.342. ✔️

Example 2: Function with a Coefficient

Problem: Express 5 tan (2x) using its cofunction.

Solution:

1. Target function: tan (2x). 2. Cofunction partner: cot. 3. Replace the angle (2x) with its complement: cot(90° − 2x).
2. Keep the coefficient 5 outside.

Answer: 5 tan (2x) = 5 cot(90° − 2x).

In radians: 5 tan (2x) = 5 cot(π⁄2 − 2x).

Example 3: Negative Angle

Problem: Rewrite − csc (π⁄6 − y) in terms of its cofunction.

Solution: 1. Target function: csc (π⁄6 − y).
2. Cofunction partner: sec.
3. Replace the angle (π⁄6 − y) with its complement: sec[π⁄2 − (π⁄6 − y)].
4. Simplify the inner expression:
π⁄2 − π⁄6 + y = (3π⁄6 − π⁄6) + y = (2π⁄6) + y = π⁄3 + y

5. Keep the negative sign outside.

Answer: −csc(π⁄6 − y) = −sec(π⁄3 + y).

Example 4: Sum of Functions

Problem: Express sin θ + cos(θ) in terms of cofunctions only.

Solution:

1. First term: sin θ → cos(90° − θ).
2. Second term: cos θ → sin(90° − θ).
3. Combine: cos(90° − θ) + sin(90° − θ).

Answer: sin θ + cos θ = cos(90° − θ) + sin(90° − θ).

Conclusion

Cofunction identities are a powerful tool for rewriting trigonometric expressions in alternative but equivalent forms. By systematically identifying the target function, swapping it for its cofunction partner, and replacing the angle with its complement, you can transform even complex expressions into more convenient or insightful forms. Mastery of these identities not only simplifies algebraic manipulation but also deepens your understanding of the symmetry inherent in trigonometry—an essential skill for solving equations, proving identities, and tackling advanced problems in mathematics and its applications.

Extending the Technique

Beyond the basic substitution, cofunction identities become especially useful when they allow you to combine or cancel terms that would otherwise remain separate. The following examples illustrate how to apply the method in slightly more involved situations while still adhering to the five‑step procedure outlined earlier.

Example 5: Multiple Angles Inside a Product

Problem: Rewrite (4\sin(3x)\cos(3x)) using only cofunctions.

Solution:

1. Identify each factor’s target function: (\sin(3x)) and (\cos(3x)).
2. Their cofunction partners are (\cos) and (\sin), respectively.
3. Replace each angle with its complement:
   (\sin(3x) \rightarrow \cos!\bigl(90^\circ-3x\bigr))
   (\cos(3x) \rightarrow \sin!\bigl(90^\circ-3x\bigr))
4. The coefficient 4 remains outside the product.
5. Combine: [ 4\sin(3x)\cos(3x)=4;\cos!\bigl(90^\circ-3x\bigr);\sin!\bigl(90^\circ-3x\bigr). ] Check (radians): Let (x=10^\circ).
   Left side: (4\sin30^\circ\cos30^\circ =4\cdot\frac12\cdot\frac{\sqrt3}{2}= \sqrt3\approx1.732).
   Right side: (4\cos60^\circ\sin60^\circ =4\cdot\frac12\cdot\frac{\sqrt3}{2}= \sqrt3). ✔️

Example 6: Nested Functions with a Negative Coefficient

Problem: Express (-2\sec\bigl(\tfrac{\pi}{4}+t\bigr)) in terms of its cofunction.

Solution: 1. Target function: (\sec\bigl(\tfrac{\pi}{4}+t\bigr)).
2. Cofunction partner: (\csc).
3. Replace the angle with its complement:
(\csc!\bigl[\tfrac{\pi}{2}-(\tfrac{\pi}{4}+t)\bigr]).
4. Simplify the inner argument:
(\tfrac{\pi}{2}-\tfrac{\pi}{4}-t = \tfrac{\pi}{4}-t).
5. Preserve the leading (-2).

Answer: (-2\sec\bigl(\tfrac{\pi}{4}+t\bigr)= -2\csc\bigl(\tfrac{\pi}{4}-t\bigr).) Verification: Choose (t=\tfrac{\pi}{6}).
Left: (-2\sec\bigl(\tfrac{\pi}{4}+\tfrac{\pi}{6}\bigr)= -2\sec\bigl(\tfrac{5\pi}{12}\bigr)\approx -2(2.414)= -4.828).
Right: (-2\csc\bigl(\tfrac{\pi}{4}-\tfrac{\pi}{6}\bigr)= -2\csc\bigl(\tfrac{\pi}{12}\bigr)\approx -2(3.864)= -7.728).
Oops – the signs differ because we missed that (\sec) and (\csc) are reciprocals of (\cos) and (\sin); the cofunction identity is (\sec\theta = \csc\bigl(\tfrac{\pi}{2}-\theta\bigr)). Applying it correctly gives:

[ -2\sec\bigl(\tfrac{\pi}{4}+t\bigr)= -2\csc\bigl(\tfrac{\pi}{2}-(\tfrac{\pi}{4}+t)\bigr)= -2\csc\bigl(\tfrac{\pi}{4}-t\bigr), ]

which matches the previous line. Re‑evaluating with (t=\tfrac{\pi}{6}):

Left: (-2\sec\bigl(\tfrac{5\pi}{12}\bigr)\approx -4.828).
Right: (-2\csc\bigl(-\tfrac{\pi}{12}\bigr)= -2\bigl[-\csc\bigl(\tfrac

These exercises highlight the versatility of the five‑step framework, allowing us to manipulate expressions systematically without losing clarity. By recognizing patterns and applying the right cofunction or reciprocal transformation, we can often simplify complex problems into manageable forms. Mastering this approach not only strengthens problem‑solving skills but also deepens our intuitive grasp of trigonometric relationships. In practice, consistency in applying each step ensures accuracy and builds confidence in tackling more challenging scenarios.

Conclusion: Utilizing the structured five‑step method enables precise rewriting of trigonometric expressions, turning seemingly complicated problems into elegant solutions while reinforcing core identities. This adaptability is invaluable for both theoretical understanding and real‑world applications.