General Sinusoid
Combine amplitude, period, phase and shift.
Key idea
Combine amplitude, period, phase and shift.
In depth
y = A sin(B(x − h)) + k describes any sinusoid. A gives amplitude, 2π/B the period, h the horizontal shift, and k the midline. To model data, find these from the maximum, minimum and period.
Worked example
Max 10, min 2, period 12: A = 4, k = 6, B = 2π/12 = π/6, so y = 4 sin(πx/6) + 6.
Formula
Real-world uses
- Fitting models to hours of daylight, tide heights and heart-rate or sensor readings
Common mistake
Using the full range rather than half for the amplitude.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is General Sinusoid?
Combine amplitude, period, phase and shift. It is a topic at the advanced level in Trigonometry.
What is the formula for General Sinusoid?
The key formula is y = A sin(B(x−C)) + D.
Where is General Sinusoid used in real life?
Real-world examples: Fitting models to hours of daylight, tide heights and heart-rate or sensor readings.
What should I learn before General Sinusoid?
Make sure you are comfortable with Vertical Shift and Midline, the previous topic in Trigonometry.
What is a common mistake with General Sinusoid?
Using the full range rather than half for the amplitude.
How can I practise General Sinusoid?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after General Sinusoid?
Move on to Graphs of Reciprocal Trig Functions next.
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.
- Graphs of Reciprocal Trig Functions – Secant, cosecant and cotangent graphs.
- Pythagorean Identity – The key identity from the unit circle.
- Other Pythagorean Identities – Divide the identity by sin² or cos².