Sketch and Use Graphs y=sin x, y=cos x, y=tan x

Sketch and Use Graphs y=sin x, y=cos x, y=tan x

Understanding the Graphs

• The graphs y = sin x, y = cos x, and y = tan x are the visual representations of the sine, cosine, and tangent functions respectively, for all values of x.
• These functions are repetitive, which means they cycle over a set period. This property makes them periodic functions.
• The period for both y = sin x and y = cos x is 2π (or 360°), while for y = tan x it is π (or 180°).

Sketching the Sine Graph (y = sin x)

• The sine graph starts at (0,0), goes to (π/2,1), then to (π,0), (3π/2,-1), and finally to (2π,0) before repeating the cycle.
• The maximum height and depth the sin graph reaches are 1 and -1 respectively.

Sketching the Cosine Graph (y = cos x)

• The cosine graph starts at (0,1), goes to (π/2,0), then to (π,-1), (3π/2,0), and finally to (2π,1) before repeating.
• The maximum height and depth the cos graph reaches are 1 and -1 respectively.

Sketching the Tangent Graph (y = tan x)

• The graph of tangent has vertical asymptotes at odd multiples of π/2 (±π/2, ±3π/2, ±5π/2, etc.) as tan x approaches positive or negative infinity.
• At multiples of π, tan x is 0.

Using the Graphs

• Periodic nature of these functions makes them useful in modelling phenomena that repeat over time like sound or light waves.
• The y = sin x and y = cos x graphs are used to represent oscillations, waveforms, or any periodic motion.
• The y = tan x graph, due to its unique shape and properties, is often used in engineering and physics to model certain types of non-repeating phenomena.

Remember:

• The graphs of these functions are useful not just in geometry, but in a variety of fields including physics, engineering, music, biology, and even finance.
• Understanding the shape, values, and properties of these graphs is key to mastering the trigonometric functions and their applications.