Graph tangent, cotangent, secant, and cosecant functions. Grade 12 Math 30-1.
Lesson 4.6 of Trigonometric Ratios & Functions in Math 30-1 — Alberta curriculum lessons.
A periodic function repeats its values in a regular pattern.
Sine and cosine are periodic functions because their graphs repeat the same shape over equal intervals.
A cycle is one complete repetition of the graph.
For both sine and cosine, one full cycle occurs over the interval:
0 ≤ x ≤ 2π
or in degrees:
0° ≤ x ≤ 360°
After 2π (or 360°), the pattern repeats again.
The graph of y = x follows a smooth wave pattern.
Key points in one cycle:.
The graph:
• starts at 0
• rises to a maximum of 1
• returns to 0
• drops to -1
• returns to 0
The graph of y = x has the same wave shape as sine but begins at its maximum value.
Key points:.
The graph:
• starts at 1
• drops to 0
• continues to -1
• rises to 0
• returns to 1
The midline is the horizontal line that is halfway between the maximum and the minimum. This line can also be called the sinusoidal axis (for both sine and cosine graphs).
The equation of the midline or sinusoidal axis is given by:
y = (max + min)/(2)
Points that lie on the midline are called midpoints. On the basic sine and cosine graphs, these are x-intercepts.
The amplitude is half the distance from the maximum to the minimum. You can also think of this as the distance between the midline and the maximum.
Amplitude is always a positive value given by:
a = |(max - min)/(2)|
For both y = x and y = x:
The cosine graph is the sine graph shifted horizontally by:
(π)/(2)
This means the graphs have the same shape, but start at different points.
A horizontal shift means every point on the graph moves the same distance left or right. When we shift sine to the left by (π)/(2), we get the cosine graph.
To see this relationship, notice that:
• (0) = 0 and ((π)/(2)) = 0
• ((π)/(2)) = 1 and (0) = 1
• (π) = 0 and ((3π)/(2)) = 0
Each point on the cosine graph appears (π)/(2) to the left of the corresponding point on the sine graph. This is why we say:
x = (x + (π)/(2))
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