Circular functions and trigonometry

• Int: Seki Takakazu calculating π to ten decimal places.
• Int: Hipparchus, Menelaus and Ptolemy.
• Int: Why are there 360 degrees in a complete turn? Links to Babylonian mathematics.
• TOK: Which is a better measure of angle: radian or degree? What are the “best” criteria by which to decide?
• TOK: Euclid’s axioms as the building blocks of Euclidean geometry. Link to non-Euclidean geometry.

• Definition of cosθ and sinθ  in terms of the unit circle.
• Definition of
• Exact values of trigonometric ratios of  and their multiples.

• The equation of a straight line through the origin is y = x tanθ
• Examples:

• Aim 8: Who really invented “Pythagoras’ theorem”?
• Int: The first work to refer explicitly to the sine as a function of an angle is the Aryabhatiya of Aryabhata (ca. 510).
• TOK: Trigonometry was developed by successive civilizations and cultures. How is mathematical knowledge considered from a sociocultural perspective?

• The Pythagorean identity cos² θ + sin² θ = 1.Double angle identities for sine and cosine.
• Relationship between trigonometric ratios.

• Simple geometrical diagrams and/or technology may be used to illustrate the double angle formulae (and other trigonometric identities).

Examples:

• Given sinθ , finding possible values of tanθ without finding θ .
• Given  and x is acute, find sin2x without finding x.

• The circular functions sin x , cos x and tan x : their domains and ranges; amplitude, their periodic nature; and their graphs.
• Composite functions of the form f (x) = asin (b(x + c) ) + d .
• Transformations.
• Applications.

• Examples:
• Example: y = sinx used to obtain y = 3sin2x by a stretch of scale factor 3 in the y-direction and a stretch of scale factor 1/2 in the x-direction.
• Link to 2.3, transformation of graphs.
• Examples include height of tide, motion of a Ferris wheel.

• Solving trigonometric equations in a finite interval, both graphically and analytically.
• Equations leading to quadratic equations in sin x, cos x or tan x

. Not required:

• the general solution of trigonometric equations.

• Solution of triangles.
• The cosine rule.
• The sine rule, including the ambiguous case.
• Area of a triangle, 
• Applications.

• Pythagoras’ theorem is a special case of the cosine rule.
• Link with 4.2, scalar product, noting that:
• Examples include navigation, problems in two and three dimensions, including angles of elevation and depression.

• Aim 8: Attributing the origin of a mathematical discovery to the wrong mathematician.
• Int: Cosine rule: Al-Kashi and Pythagoras.
• TOK: Non-Euclidean geometry: angle sum on a globe greater than 180°.