Trigonometric functions and circular measure - the Australian ...

A guide for teachers – Years 11 and 12 • {19}Exercise 5A triangle has two sides of length 5 cm and 4 cm containing an angle θ. Its area is 5 cm 2 .Find the two possible (exact) values of θ and draw the two triangles that satisfy the giveninformation.Exercise 6Write down two different expressions for the area of a triangle ABC and derive the sinerule from them.Trigonometric identitiesThe Pythagorean identityThere are many important relationships between the trigonometric functions which areof great use, especially in calculus. The most fundamental of these is the Pythagoreanidentity. For acute angles, this is easily proven from the following triangle ABC withhypotenuse of unit length.B1sin θAθcos θCWith ∠B AC = θ, we see that AC = cosθ and BC = sinθ. Hence Pythagoras’ theorem tellsus thatcos 2 θ + sin 2 θ = 1.This formula holds for all angles, since every angle can be related to an angle in the firstquadrant whose sines and cosines differ only by a sign, which is dealt with by squaring.Dividing this equation respectively by cos 2 θ and by sin 2 θ, we obtain1 + tan 2 θ = sec 2 θ and cot 2 θ + 1 = cosec 2 θ.From these standard identities, we can prove a variety of results.