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3.3 Trigonometry

Syllabus
9709–2028–2029
Topic
3.3
Level
A2

Learning objectives

3.3.1Reciprocal trig functions• understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude3.3.2Trig identities• use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, and select an identity or identities appropriate to the context, showing familiarity in particular with the use of - sect an122 /ii + and cosecc ot122 /ii + - the expansions of sin(A ± B), cos(A ± B) and tan(A ± B) - the formulae for sin 2A, cos 2A and tan 2A - the expression of sinc osab ii+ in the forms sinR ! ai^h and cosR ! ai^h. e.g. simplifying cos(x - 30˚) - 3 sin(x - 60˚). e.g. solving tanc ot 4ii+ =, sect an25 2 -ii =, coss in32 1ii+ =.

Build reciprocal trig graphs from denominator behaviour

\sec x= rac1{\cos x},\qquad \cosec x= rac1{\sin x},\qquad \cot x= rac1{ an x}= rac{\cos x}{\sin x},whereverthedenominatorisnon−zero.wherever the denominator is non-zero.

Function Period Range Vertical asymptotes
sec⁡x\sec x 2π2\pi y≤−1y\le-1 or y≥1y\ge1 cos⁡x=0\cos x=0
cosec⁡x\cosec x 2π2\pi y≤−1y\le-1 or y≥1y\ge1 sin⁡x=0\sin x=0
cot⁡x\cot x π\pi all real yy sin⁡x=0\sin x=0

Start with the corresponding cosine, sine or tangent graph over the required angles. Its denominator zeros become reciprocal asymptotes; where the denominator is ±1\pm1, the reciprocal is also ±1\pm1; use the denominator sign between asymptotes to choose each branch.

sec⁡(π/3)=2\sec(\pi/3)=2 because cos⁡(π/3)=1/2\cos(\pi/3)=1/2. At x=π/2x=\pi/2, cosine is zero, so secant is undefined and has a vertical asymptote.

Reciprocal functions have no zeros: 1/f(x)1/f(x) cannot equal 00. cosec⁡x\cosec x is not sin⁡−1x\sin^{-1}x; inverse notation means a principal angle, not a reciprocal.

Select the identity family that exposes the required form

1+ an^2A=\sec^2A,\qquad 1+\cot^2A=\cosec^2A.Usethesetoexchangeareciprocalsquareforatangent/cotangentsquare.Use these to exchange a reciprocal square for a tangent/cotangent square.

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin B,\cos(A\pm B)=\cos A\cos B\mp\sin A\sin B, an(A\pm B)= rac{ an A\pm an B}{1\mp an A an B}.

\sin2A=2\sin A\cos A,\quad \cos2A=\cos^2A-\sin^2A=1-2\sin^2A=2\cos^2A-1, an2A= rac{2 an A}{1- an^2A}.

Write asin⁡heta+bcos⁡heta=Rsin⁡(heta+α)a\sin heta+b\cos heta=R\sin( heta+\alpha) by matching Rcos⁡α=aR\cos\alpha=a and Rsin⁡α=bR\sin\alpha=b. Thus R=a2+b2R=\sqrt{a^2+b^2} and choose the quadrant of α\alpha from both signs. A cosine form is equally valid when coefficient matching is adjusted.

For exact values, expand a compound angle built from known angles. For equations, rewrite to one function or RR-form, use its range to decide existence, find all solutions in the stated interval, and check any denominator restrictions.

The sign in the cosine compound formula is opposite the sign between the angles. Identity manipulation does not remove excluded values introduced by dividing by sine, cosine or another expression.