Reciprocal Trig Functions: csc, sec, cot

Trigonometry — The Other Three Trig Functions

In this lesson:

• Define csc, sec, and cot as reciprocals
• Evaluate them for standard angles
• Simplify expressions and solve ACT problems

What You Will Learn Today

After this lesson, you will be able to:

1. Define csc, sec, cot as reciprocals of sin, cos, tan
2. Express them as right-triangle side ratios
3. Evaluate them for 30°, 45°, and 60°
4. Simplify expressions with reciprocal trig functions
5. Solve ACT-style problems efficiently

You Already Know Three Trig Functions

SOH-CAH-TOA gives us three ratios:

But there are six trig functions total. What are the other three?

Meet the Three Reciprocal Functions

Each trig function has a reciprocal partner:

Memory Aid for Reciprocal Pairs

Pairs with the function that does NOT share its first letter:

The letters deliberately cross over.

Right Triangle With All Six Ratios

The reciprocal ratios use the same sides, just flipped.

Quick Check on Reciprocal Pairs

Answer these without looking back:

1. What is the reciprocal of ?
2. What is the reciprocal of ?

Think about it before advancing...

The Method for Evaluating Reciprocals

To find a reciprocal trig value:

Step 1: Find the base trig value (sin, cos, or tan)

Step 2: Flip it — take the reciprocal

Step 3: Rationalize the denominator if needed

The angle stays the same. You flip the output, not the input.

Worked Example: Find csc 30 Degrees

Step 1: Find the base value

Step 2: Flip it

No rationalization needed — the answer is 2.

Worked Example: Find sec 45 Degrees

Step 1: Find the base value

Step 2: Flip and rationalize

Worked Example: Find cot 60 Degrees

Method 1: Flip tangent

Method 2: Use

Quick Check: Find csc 45 Degrees

What is ?

Flip it, then rationalize if needed...

Reference Table for Standard Angles

Note: csc and sec are undefined where sin or cos = 0.

Reciprocal Products Always Equal One

A function times its reciprocal always equals 1:

This is the key identity for ACT simplification.

Worked Example: Simplify a Reciprocal Expression

Simplify:

Step 1: Recognize reciprocal pairs

Step 2: Add the results

Worked Example: Simplify cot Times sin

Simplify:

Strategy: Convert to sin and cos first

The cancels, leaving just .

ACT Practice: Evaluate csc 60 Degrees

What is ?

A.   B.   C.   D.   E.

Solve it, then advance...

Answer: , so → C

ACT Practice: Simplify a Trig Expression

Simplify:

A.   B.   C.   D.   E.

Identify the reciprocal pairs...

Answer: → B

Key Takeaways and Common Mistakes

Remember:

• , ,
• Pairs cross: sincsc, cossec, tancot
• Convert to sin/cos when simplifying

Watch out:

• sin pairs with csc, NOT sec
• (that's arcsin)
• Always rationalize — ACT answers never have in denominators

Coming Up Next in Trigonometry

Up next: Trigonometric identities and equations

• Pythagorean identities:
• Solving equations with reciprocal trig functions
• Connecting all six functions in complex problems