Trigonometry

Builds on

Required prerequisites, all the way down: 18

You will learn to

• Define sine, cosine and tangent for acute angles via similar right triangles, and for all angles via the unit circle.
• Convert between degrees and radians and sketch the graphs of sin, cos and tan.
• Prove and apply sin²θ + cos²θ = 1, the addition formulas, and the laws of sines and cosines.

Definition

• For an acute angle θ of a right triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent.
• On the unit circle, the point at angle θ measured anticlockwise from the positive x-axis is (cos θ, sin θ).
• An angle of one radian subtends an arc equal in length to the radius; 2π radians equal 360°.

Theorem

• sin²θ + cos²θ = 1.
• sin(α + β) = sin α cos β + cos α sin β and cos(α + β) = cos α cos β − sin α sin β.
• Law of cosines: c² = a² + b² − 2ab cos C.
• Law of sines: a/sin A = b/sin B = c/sin C.

Common misconceptions

• Reading sin θ as 'sin times θ'.
• Using degree mode when radians are required, or vice versa.
• Thinking sin(α + β) = sin α + sin β.

Leads to

Optional discoveries

• Partial differential equations and Fourier series Periodic signals can be built from sines and cosines: the idea behind Fourier series.

Learning pathways

• Conventional K–12 progression
• Accelerated secondary progression

Further reading

• I. M. Gelfand and Mark Saul, Trigonometry (2001)