Preparing for University Calculus - Math and Computer Science

3.13 Trigonometric identitiesThere are many identities that are satisfied by the trigonometric functions. Theyare important in calculus because they are used to reduce the number of rules youhave to learn. Note that cos 2 (α) means (cos(α)) 2 .The identities for one angle can all be derived from the definitions of the six functions,via the relations tan(α) = sin(α)/ cos(α), cot(α) = cos(α)/ sin(α), sec(α) =1/ cos(α) csc(α) =1/ sin(α) and from the identities sin 2 (α)+cos 2 (α) = 1, sec 2 (α)−tan 2 (α) = 1, and csc 2 (α) − cot 2 (α) =1.There are also various identities for trigonometric functions of sums and differencesof angles, and for double and half angles. These are aso useful in calculus.Examples:1. sin 2 (θ) sec(θ) csc(θ) =(a) sin(θ) (b) cos(θ) (c) tan(θ) (d) sec(θ) (e) csc(θ) (f) cot(θ)Solution:sin 2 (θ) sec(θ) csc(θ) = (sin(θ) sec(θ)) (sin(θ) csc(θ))= (sin(θ)(1/ cos(θ))) (sin(θ)(1/ sin(θ)))= (tan(θ))so the answer is (c).2. If sin(35 ◦ ) = cos(β), and β ∈ [0 ◦ , 90 ◦ ], find β.Solution: sin(α) = cos(90 ◦ − α) for all α; so here β =55 ◦ .3. If cos(α) =0.3, find cos(2α).Solution: We do not need to know α for this! One form of the “double angleformula” for cosines iscos(2α) = 2 cos 2 (α) − 1from which we see that cos(2α) = 2(0.3 2 ) − 1=−0.82.34