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Maths A-level: TrigonometryIdentitiesprint page Bookmark and Share With Your Friends!Graphs of sec x, cosec x, cot xYou will also need to know the graphs and properties of thereciprocal functions:The following properties apply to any reciprocal function: 1. The reciprocal of zero is +∞ 2. The reciprocal of +∞ is zero 3. The reciprocal of 1 is 1 4. The reciprocal of -1 is -1 5. Where the function has a maximum value, its reciprocal has a minimum value 6. If a function increases, the reciprocal decreases 7. A function and its reciprocal have the same signThe curves of cosec x, sec x and cot x are shown below:
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From a right angled triangle we know that:cos2θ + sin2θ = 1It can also be shown that:1 + tan2θ = sec2θ and cot2θ + 1 = cosec2θ(Try dividing the second expression by cos2θ to get the first rearrangement,and separately divide cos2θ + sin2θ = 1, by sin2θ to get the other formula.)These are Trigonometric Identities and useful for rewriting equations sothat they can be solved, integrated, simplified etc.Formulae for sin (A + B), cos (A + B), tan (A + B)Trigonometric functions of angles like A + B and A − B can be expressed interms of the trigonometric functions of A and B.These are called compound angle identities:sin (A + B) = sin A cos B + cos A sin Bsin (A - B) = sin A cos B - cos A sin Bcos (A + B) = cos A cos B - sin A sin B
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cos (A - B) = cos A cos B + sin A sin BRemember: take care with the signs when using these formulae.Double angle formulaeThe compound angle formulae can also be used with two equal angles i.e. A= B.If we replace B with A in the compound angle formulae for (A + B),we have:sin 2A = 2(sin A cos A)cos 2A = cos2A - sin2AAlso,cos 2A = cos 2A - sin 2A = 1 - 2sin2A = 2cos2A - 1The use for this final rearrangement is when integrating cos2x orsin2x.We use cos2 x = ½cos 2x + ½ and sin2 x = ½ - ½ cos 2x which we canintegrate.Half angle formulaeUsing this double angle formula for tan 2A and the two identities:We can replace 2A with x and use T for tan(x/2).This gives us the following identities, which allow all the trigonometricfunctions of any angle to be expressed in terms of T.Factor formulaeThe formulae we have met so far involve manipulating single expressions ofsin x and cos x. If we wish to add sin or cos expressions together we
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need to use the factor formulae, which are derived from the compoundangle rules we met earlier.The compound angle formulae can be combined to give:2sin A cos B = sin (A + B) + sin (A − B) 2cos A sin B = sin (A + B) - sin (A − B)2cos A cos B = cos (A + B) + cos (A − B)−2sin A sin B = cos (A + B) - cos (A − B)If we simplify the right hand side of each of these equations bysubstitutingA + B = J and A − B = K, we create the factor formulae:The "Rcos" functionThe factor formulae allow us to add and subtract expressions that are allsines or all cosines. If we wish to add a sine and a cosine expression togetherwe have to use a different method.This method is based upon the fact that combining a sine and a cosine willgenerate another cos curve with a greater amplitude and which is anumber of degrees out of phase with the graph of cos θ.This means that it can be written as R cos(θ - α), where R represents theamplitude and α represents the number of degrees the graph is out of phase(to the right).The solution is based upon the expansion of cos(θ - α).Example:Write 5 sin x + 12 cos x in the form R cos (θ - α)R cos (θ - α) = R (cos θ cos α + sin θ sin α)By matching this expansion to the question we get:
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R cos θ cos α = 12 cos θ and R sin θ sin α = 5 sin θThis gives:R cos α = 12 and R sin α = 5By illustrating this with a right-angled triangle, we get,Therefore: α = 22.6oTherefore: 5 sin θ + 12 cos θ = 13 cos(θ - 22.6)It has a maximum value of 13 and is 22.6o out of phase with the graph of cosθ.Note: This procedure would work with Rsin(θ + α).Check to see if you can get a similar answer - it should be 13 sin (θ + 67.4)
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