Graphs of Trigonometric Functions
This topic covers the fundamental properties of the sine, cosine, and tangent functions, focusing on their 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.