Graphs of Reciprocal Trig Functions

Key idea

Secant, cosecant and cotangent graphs.

In depth

csc x = 1/sin x has vertical asymptotes wherever sin x = 0, and its branches touch ±1 where sine does. sec x does the same with cosine; cot x mirrors tangent.

Worked example

csc x has minimum value 1 at x = π/2 and asymptotes at 0, π, 2π.

Formula

Real-world uses

• Analysing functions in signal processing and engineering control systems

Common mistake

Forgetting that reciprocal graphs never lie between −1 and 1.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Trigonometry

• Graph of Tangent – Periodic with vertical asymptotes at odd multiples of π/2.
• Phase Shift – Horizontal translation of a wave.
• Vertical Shift and Midline – Moves the wave up or down.
• General Sinusoid – Combine amplitude, period, phase and shift.
• Pythagorean Identity – The key identity from the unit circle.
• Other Pythagorean Identities – Divide the identity by sin² or cos².