Algebra 2
Trigonometric functions (introduction)
Graphs of sine, cosine, and tangent
20 practice questions
0 video lessons
Theory + worked examples
Theory
Sine and cosine are periodic waves. For \(y=a\sin(bx)\):
- Amplitude \(=|a|\) — the height from the midline.
- Period \(=\dfrac{2\pi}{b}\) — one full cycle.
- Their range is \([-1,1]\) before scaling.
Tangent has period \(\pi\) and vertical asymptotes where cosine is \(0\).
Sine and cosine: amplitude 1, period \(2\pi\).
Reading a sinusoid.
Amplitude and period:
\[y=a\sin(bx):\quad \text{amplitude}=|a|,\ \text{period}=\dfrac{2\pi}{b}\]
Cosine leads sine by a quarter period.
How to read a trig graph
- Read \(|a|\) for the amplitude.
- Compute \(\dfrac{2\pi}{b}\) for the period.
- Identify the midline and range.
- For tangent, mark the asymptotes.
Example 1 — Amplitude and period
State the amplitude and period of \(y=\sin x\).
Solution
Read the base wave.
| \(\text{amplitude}\) | \(=\) | \(1\) |
| \(\text{period}\) | \(=\) | \(2\pi\) |
Example 2 — A stretched wave
State the amplitude of \(y=3\sin x\).
Solution
The coefficient sets the amplitude.
| \(\text{amplitude}\) | \(=\) | \(|3|=3\) |
Example 3 — A shorter period
Find the period of \(y=\sin(2x)\).
Solution
Use period \(=\dfrac{2\pi}{b}\).
| \(\text{period}\) | \(=\) | \(\dfrac{2\pi}{2}=\pi\) |
Example 4 — Maximum of cosine
Where does \(y=\cos x\) reach its maximum on \([0,2\pi]\)?
Solution
Cosine starts at its maximum.
| \(\cos 0\) | \(=\) | \(1\ \text{(max at }x=0)\) |
Common pitfalls
Period is \(\dfrac{2\pi}{b}\), not \(2\pi b\).
Amplitude is \(|a|\), always positive.
Tangent has asymptotes, unlike sine and cosine.
Frequently asked questions
What is the amplitude of a sinusoid?
\(|a|\), the height from the midline to a peak.
What is the period of \(y=\\sin(bx)\)?
\(\dfrac{2\pi}{b}\).
What is the range of sine and cosine?
\([-1,1]\) before any vertical stretch.
How is the tangent graph different?
It has period \(\pi\) and vertical asymptotes.
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