Cofunction Identities: Step-by-Step Guide
Cofunction identities connect the trig functions of two angles that add up to 90 degrees (pi/2). The pairs are sine and cosine, tangent and cotangent, and secant and cosecant. The main one is sin(pi/2 - x) = cos(x). They work because the two acute angles in a right triangle add up to 90 degrees, so the 'opposite' side for one angle is the 'adjacent' side for the other.
Math Racer skill guide · Trigonometry · Grades 11-12
How to solve it, step by step
- Find the cofunction pair. Sin goes with cos, tan goes with cot, and sec goes with csc.\[(\sin,\cos),(\tan,\cot),(\sec,\csc)\]
- Use the identity. Replace f(pi/2 - x) with the cofunction of x.\[\sin(\tfrac{\pi}{2} - x) = \cos x\]
- Work out the angle if you can. Subtract pi/2 - x if the angle is a number.\[\tfrac{\pi}{2} - x\]
- Find the value. Use the unit circle or another identity.\[\cos(\theta) = \text{value}\]
- Check. Make sure the original and the final expression give the same number.\[\sin\theta = \cos(\tfrac{\pi}{2}-\theta)\]
Worked example
Simplify sin(pi/2 - pi/3).
Step 1: Sine and cosine are cofunctions.
Step 2: sin(pi/2 - x) = cos(x), so sin(pi/2 - pi/3) = cos(pi/3).
Step 3: As a check, pi/2 - pi/3 = pi/6.
Step 4: cos(pi/3) = 1/2. Also sin(pi/6) = 1/2.
Step 5: Both ways give 1/2. It works.
The answer is 1/2.
Tips and common mistakes
- All six: sin and cos, tan and cot, sec and csc swap when the angle becomes pi/2 - x.
- In degrees: sin(90 degrees - x) = cos(x), and so on.
- These identities exist because the two acute angles of a right triangle add up to 90 degrees.
- Use them to rewrite an expression in terms of one trig function so it is easier to simplify.
- Together with the even and odd rules, they handle most angle reductions.
- They are also useful in calculus when working with trig expressions.
- tan(pi/2 - x) = cot(x). Watch for places where cotangent is undefined.
- sec(pi/2 - x) = csc(x). Reciprocals stay paired.