Basic Trigonometry

Basic Trigonometry in Pure Mathematics

Definitions and Basic Concepts

• Understand, remember, and apply the relationship between the sides and angles of a right-angled triangle as defined by the sine (sin), cosine (cos), and tangent (tan) ratios.
• Familiarise yourself with the unit circle and use it to relate angles and points on a coordinate plane, to define the sin, cos, and tan of an angle.
• Recall and use the reciprocal trigonometric ratios - cosecant (csc), secant (sec), and cotangent (cot), defined respectively as the reciprocals of sin, cos, and tan.

Trigonometric Identities

• Apply the Pythagorean identities: sin²θ + cos²θ = 1, 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ.
• Use double-angle and half-angle identities to simplify and solve trigonometric expressions.
• Be familiar with the sum and difference formulas for sin, cos, and tan.

Solving Trigonometric Equations

• Demonstrate fluency in solving trigonometric equations for a desired range, including those involving multiple angles or compound angles.
• Know how to solve trigonometric equations resulting in quadratic forms, such as sin²θ = cos²θ.

Graphs of Trigonometric Functions

• Be able to draw and interpret graphs of y = sin x, y = cos x, and y = tan x.
• Understand the impact of transformations such as translations, reflections, and stretches on these graphs.
• Be able to identify and use period, amplitude, and phase shift of trigonometric functions from their graphs.