Write The Following Function In Terms Of Its Cofunction

Write the following function in terms of its cofunction is a common request when students first encounter trigonometric identities. Mastering this skill not only simplifies expressions but also reveals the deep symmetry between complementary angles. In this guide we will explore the theory behind cofunction identities, walk through a clear step‑by‑step method, provide numerous worked examples, discuss practical applications, and answer frequently asked questions. By the end you’ll feel confident converting any of the six basic trigonometric functions into its cofunction counterpart.

Understanding Cofunction Identities

Cofunction identities relate the value of a trigonometric function of an angle to the value of its cofunction evaluated at the complementary angle (i.e., (90^\circ - \theta) or (\frac{\pi}{2} - \theta) in radians). The six basic functions pair up as follows:

These relationships stem from the geometry of a right triangle: the side opposite one acute angle is adjacent to the other, and the hypotenuse is shared. Consequently, the sine of one angle equals the cosine of its complement, and similarly for the other pairs.

Key takeaway: To rewrite a function in terms of its cofunction, replace the angle (\theta) with its complement (\frac{\pi}{2}-\theta) and swap the function name with its paired cofunction.

How to Write a Function in Terms of Its Cofunction – Step‑by‑Step

Follow this systematic procedure for any trigonometric expression:

1. Identify the target function
   Determine which of the six basic functions you need to convert (e.g., (\sin), (\tan), (\sec)).

2. Recall its cofunction pair
   Use the table above to find the matching cofunction (e.g., (\sin \leftrightarrow \cos)).

3. Replace the angle with its complement
   Substitute (\theta) by (\frac{\pi}{2}-\theta) (or (90^\circ-\theta) if working in degrees).

4. Write the new expression
   Place the cofunction name in front of the complemented angle.

5. Simplify if necessary Apply algebraic rules, distribute negatives, or combine like terms.
   Tip: Keep the angle inside the parentheses; you do not need to expand (\frac{\pi}{2}-\theta) unless the problem specifically asks for an expanded form.

6. Verify (optional)
   Plug a sample angle (e.g., (\theta = 30^\circ)) into both the original and transformed expressions to confirm equality.

Bold reminder: Never change the sign of the angle unless the identity itself introduces a negative (which occurs only for certain shifted identities, not the basic cofunction formulas).

Examples of Cofunction Transformations

Example 1: Simple Sine to Cosine

Problem: Write (\sin\left(\frac{\pi}{6}\right)) in terms of its cofunction.

Solution:

1. Target function: (\sin).
2. Cofunction pair: (\cos).
3. Complement angle: (\frac{\pi}{2} - \frac{\pi}{6} = \frac{\pi}{3}). 4. New expression: (\cos!\left(\frac{\pi}{3}\right)). Thus, (\displaystyle \sin\left(\frac{\pi}{6}\right)=\cos!\left(\frac{\pi}{3}\right)).
   Check: (\sin 30^\circ = 0.5) and (\cos 60^\circ = 0.5).

Example 2: Tangent to Cotangent with a Variable

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

Solution:

1. Target: (\tan).
2. Cofunction: (\cot).
3. Complement: (\frac{\pi}{2} - 2x).
4. Result: (\displaystyle \tan(2x)=\cot!\left(\frac{\pi}{2}-2x\right)).

No further simplification is needed unless the context demands expanding the argument.

Example 3: Secant to Cosecant with a Negative Angle

Problem: Rewrite (\sec(-\theta)) in terms of its cofunction.

Solution:

1. Target: (\sec).
2. Cofunction: (\csc).
3. Complement: (\frac{\pi}{2} - (-\theta) = \frac{\pi}{2} + \theta).
4. Intermediate: (\sec(-\theta)=\csc!\left(\frac{\pi}{2}+\theta\right)). Recall that (\sec) is an even function: (\sec(-\theta)=\sec\theta). If desired, you could also write (\sec\theta=\csc!\left(\frac{\pi}{2}-\theta\right)); both forms are correct because adding (\pi) to the argument of cosecant flips its sign, which is compensated by the evenness of secant.

Example 4: A Composite Expression

Problem: Convert (\displaystyle \frac{\sin x}{\cos x}) into cofunctions only.

Solution:

• Recognize the fraction as (\tan x).
• Apply the tangent‑to‑cotangent identity: (\tan x = \cot!\left(\frac{\pi}{2}-x\right)).

Thus, (\displaystyle \frac{\sin x}{\cos x}= \cot!\left(\frac{\pi}{2}-x\right)).

If the instruction demands only cofunctions (no tangent), this is the final form.

Why Cofunction Identities Matter – Applications

1. Simplifying Integrals and Derivatives