Graphs of Trigonometric Functions
The sine, cosine, and tangent functions have characteristic graphical shapes, repeating patterns (periodicity), and symmetries; mastering these is crucial for solving trigonometric equations and analysing wave-like phenomena without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The graphs of sin(x) and cos(x) are waves (sinusoids) with an amplitude of 1 and a period of 2π. They are identical in shape, but cos(x) is horizontally shifted by π/2 relative to sin(x).
- The graph of tan(x) is periodic with a period of π. It has vertical asymptotes wherever cos(x) = 0 and its range is all real numbers.e.g., at x = π/2, 3π/2
- Symmetry is key: cos(x) is an even function (cos(-x) = cos(x)), symmetrical about the y-axis. sin(x) and tan(x) are odd functions (sin(-x) = -sin(x)), with 180° rotational symmetry about the origin.
- For functions like y = A sin(Bx), '|A|' is the amplitude (vertical stretch) and the period is 2π/|B| (horizontal stretch/compression). For y = A tan(Bx), the period is π/|B|.
Diagram
Formulae
tan(x) = sin(x) / cos(x) To relate the three basic trig functions, find values for tan(x), or identify the vertical asymptotes of tan(x) by finding where cos(x) = 0.
Period of sin(Bx) or cos(Bx) = 2*pi / |B| To determine the new period of a sine or cosine function after a horizontal stretch or compression.
Period of tan(Bx) = pi / |B| To determine the new period of a tangent function after a horizontal stretch or compression.
Definitions
- Period
- The length of the smallest horizontal interval over which the function's graph repeats. For sin(x) and cos(x) the period is 2π; for tan(x) it is π.
- Amplitude
- For sinusoidal functions like sin(x) and cos(x), it is half the vertical distance between the maximum and minimum values. It measures the 'height' of the wave from its central axis.
- Asymptote
- A line that a curve approaches but never touches. The graph of tan(x) has vertical asymptotes at x = (n + 1/2)π for any integer n.
Worked example
Consider the function f(x) = 3 - 2sin(x) over the domain 0 ≤ x ≤ 2π. How many solutions does the equation f(x) = 2 have in this domain?
- 1
Set up the equation:
3 - 2sin(x) = 2 - 2
Rearrange the equation to isolate the sin(x) term.
Subtracting 3 from both sides gives -2sin(x) = -1.
- 3
Solve for sin(x):
Divide by -2 to get sin(x) = 1/2.
- 4
Recall the principal value for sin(x) = 1/2.
This is x = π/6 - 5
Consider the graph of sin(x) or use the unit circle to find all solutions in the domain 0 ≤ x ≤ 2π.
The sine function is positive in the first and second quadrants.
- 6
The first quadrant solution is x = π/6.
- 7
The second quadrant solution is found using symmetry:
x = π - π/6 = 5π/6 - 8
Both solutions, π/6 and 5π/6, are within the given domain.
- 9
Therefore, there are two distinct solutions.
Answer: 2
Common mistakes
- ×Forgetting the vertical asymptotes for y = tan(x). Its domain is not all real numbers, and it is undefined at odd multiples of π/2. This is a common source of domain confusion.
- ×Mixing up the periods. The period of tan(x) is π, while the period for both sin(x) and cos(x) is 2π. Applying transformations like sin(2x) halves the period to π.
- ×Misinterpreting symmetry. Confusing even (cos(x)) and odd (sin(x), tan(x)) functions can lead to incorrect sign calculations, for example when evaluating trig functions for negative angles.
No-calculator tips
- ✓Quickly sketch the basic shapes of sin, cos, and tan from memory. Mark key points at x = 0, π/2, π, 3π/2, 2π to guide your sketch and any transformations.
- ✓Memorise the exact trig values for 0, π/6, π/4, π/3, and π/2. Use the 'special triangles' (the isosceles right-angled triangle and half an equilateral triangle) to derive them if you forget.
- ✓Use the CAST diagram (or just picture the unit circle) to rapidly determine the sign (+ or -) of each trig function in the four quadrants, which is essential for finding all solutions to an equation within a given range.