4-4 The trigonometric functions of any angle

• Take a point P(x, y) on the terminal side at distance r = √(x² + y²) from the origin: sinθ = y/r, cosθ = x/r, tanθ = y/x (x ≠ 0).
• Reciprocals: cscθ = r/y, secθ = r/x, cotθ = x/y — csc is the reciprocal of sin, sec of cos, cot of tan.
• tanθ = sinθ/cosθ and cotθ = cosθ/sinθ; cotθ is undefined when sinθ = 0, tanθ and secθ when cosθ = 0.

Signs in the quadrants

• First: all six positive. Second: only sin (and csc) positive. Third: only tan (and cot) positive. Fourth: only cos (and sec) positive.
• Mnemonic: C AST — All, Sin, Tan, Cos going round the quadrants counter-clockwise.

Quadrantal values

• 0°: sin 0, cos 1, tan 0. 90°: sin 1, cos 0, tan undefined. 180°: sin 0, cos −1, tan 0. 270°: sin −1, cos 0, tan undefined. 360°: as 0°.

Special acute angles

• 30°: sin = 1/2, cos = √3/2, tan = 1/√3. 45°: sin = cos = √2/2, tan = 1. 60°: sin = √3/2, cos = 1/2, tan = √3.
• The 3-4-5 approximations used in the syllabus: sin 37° ≈ 0.6, cos 37° ≈ 0.8, tan 37° ≈ 0.75; sin 53° ≈ 0.8, cos 53° ≈ 0.6, tan 53° ≈ 4/3.