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 by rewriting in sine and cosine, and solve right-triangle problems

What You Will Learn Today

1. Define csc, sec, and cot from sin, cos, and tan
2. Write them as side ratios: hyp/opp, hyp/adj, adj/opp
3. Evaluate at 30°, 45°, and 60°, rationalizing
4. Simplify by rewriting in sin and cos, even without reciprocal pairs
5. Solve right-triangle problems using a reciprocal ratio

Six Trig Functions, but You Know Three

• What are the other three?

Meet the Three Reciprocal Functions

In the 3-4-5 triangle: , , .

Reciprocal Pairs Never Share a First Letter

is a number; is an angle.

Right Triangle With All Six Ratios

The reciprocal ratios use the same sides, just flipped.

Flip the Output, Not the Angle

To find csc 30°: Step 1, the base value; Step 2, flip it; Step 3, rationalize if needed.

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

Reference Table for Standard Angles

The test calculator disables csc, sec, and cot: .

ACT Practice: Evaluate csc 60 Degrees

What is ?

(a)   (b)   (c)   (d)

Answer: , so , choice (c).

Simplify: A Function Times Its Reciprocal

A function times its reciprocal partner is .

Not a Pair? Rewrite in Sine and Cosine

Recall . Sin pairs with csc, not sec, so . Replace tan, cot, sec, and csc with sin and cos, then cancel.

Right-Triangle Problem With a Given Secant

, and the side adjacent to is . Find the hypotenuse.

Key Takeaways and Common Mistakes

• , ,
• A reciprocal ratio is a flipped side ratio
• Pair: product is 1. Else rewrite in sin and cos

Watch out:

• sin pairs with csc, NOT sec
• (arcsin)
• Flip the output, not the angle
• Rationalize every answer

Coming Up Next in Trigonometry

Up next: Identities built on today's rewrite

• Rewriting in sine and cosine leads to the Pythagorean identity
• The unit circle shows where csc, sec, and cot are undefined