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Trigonometric identities
The Pythagorean identity
There are many important relationships between the trigonometric functions which are of great use, especially in calculus. The most fundamental of these is the Pythagorean identity. For acute angles, this is easily proven from the following triangle ABC with hypotenuse of unit length.
Detailed description of diagram
With ∠BAC=θ, we see that AC=cosθ and BC=sinθ. Hence Pythagoras' theorem tells us that
cos2θ+sin2θ=1.This formula holds for all angles, since every angle can be related to an angle in the first quadrant whose sines and cosines differ only by a sign, which is dealt with by squaring. Dividing this equation respectively by cos2θ and by sin2θ, we obtain
1+tan2θ=sec2θandcot2θ+1=cosec2θ.From these standard identities, we can prove a variety of results.
Example
Prove the following identities:
- (1−sinθ)(1+sinθ)=cos2θ
- 2cos3θ−cosθsinθcos2θ−sin3θ=cotθ.
Solution
- LHS=(1−sinθ)(1+sinθ)=1−sin2θ(difference of two squares)=cos2θ(Pythagorean identity)=RHS
- LHS=2cos3θ−cosθsinθcos2θ−sin3θ=cosθ(2cos2θ−1)sinθ(cos2θ−sin2θ)=cosθ(2cos2θ−1)sinθ(cos2θ−(1−cos2θ))=cosθ(2cos2θ−1)sinθ(2cos2θ−1)=cosθsinθ=cotθ=RHS
Exercise 7
Prove that
1secθ+tanθ=cosθ1+sinθ.Double angle formulas
The sine and cosine functions are not linear. For example, cos(A+B)≠cosA+cosB. The correct formula for cos(A+B) is given in the next subsection. In the special case when A=B=θ, we would like to obtain a simple formula for cos(2θ)=cos2θ, called the double angle formula. It is particularly useful in applications and in calculus and is quite easy to derive.
Consider the following isosceles triangle ABC, with sides AB and AC both of length 1 and apex angle 2θ. We will need to assume, for the present, that 0∘<θ<90∘.
Let AD be the perpendicular bisector of the interval BC. Then, using basic properties of an isosceles triangle, we know that AD bisects ∠BAC and so BD=DC=sinθ. Applying the cosine rule to the triangle ABC, we immediately have
cos2θ=12+12−4sin2θ2×1×1=2−4sin2θ2=1−2sin2θ.Replacing 1 by cos2θ+sin2θ, we arrive at the double angle formula
cos2θ=cos2θ−sin2θ.This may also be written as cos2θ=2cos2θ−1 or cos2θ=1−2sin2θ, but is best learnt in the form given above.
We derived this formula under the assumption that θ was between 0∘ and 90∘, but the formula in fact holds for all values of θ. We shall see this in the next subsection, where we prove the general expansion formula for cos(A+B).
Example
Find cos2212∘ in surd form.
Solution
Putting θ=2212∘ into the double angle formula cos2θ=2cos2θ−1 and writing x for cos2212∘, we obtain
cos45∘=2x2−11√2=2x2−12x2=1+√2√2x=√1+√22√2, which simplifies to 12√2+√2.We can also find a double angle formula for sine using the same diagram.
In this case, we write down formulas for the area of △ABC in two ways:
- On the one hand, the area is given by 12AB⋅ACsin(∠BAC)=12sin2θ.
- Since AD=cosθ, we can alternatively split the triangle into two right-angled triangles and write the area as 2×12sinθcosθ=sinθcosθ.
Equating these two expressions for the area, we obtain
sin2θ=2sinθcosθ.As before, we assumed that θ lies between 0∘ and 90∘, but the formula is valid for all values of θ.
Exercise 8
Find sin15∘ in surd form.
From the Pythagorean identity and the double angle formula for cosine, we have
cos2θ+sin2θ=1cos2θ−sin2θ=cos2θ.Adding these equations and dividing by 2 we obtain cos2θ=12(cos2θ+1), while subtracting them and dividing by 2 we obtain sin2θ=12(1−cos2θ). These formulas are very important in integral calculus, as discussed in the module The calculus of trigonometric functions.
Exercise 9
Use the double angle formulas for sine and cosine to show that
tan2θ=2tanθ1−tan2θ,for tanθ≠±1.sin2θ=2sinθcosθ | |
cos2θ=cos2θ−sin2θ=2cos2θ−1=1−2sin2θ | |
tan2θ=2tanθ1−tan2θ, | for tanθ≠±1 |
Trigonometric functions of compound angles
In the previous subsection, we derived formulas for the trigonometric functions of double angles. That derivation, which used triangles, was only valid for a limited range of angles, although the formulas remain true for all angles.
In this subsection we find the expansion formulas for sin(A+B), sin(A−B), cos(A+B) and cos(A−B), which are valid for all A and B. The double angle formulas can be recovered by putting A=B=θ.
These formulas are quite central in trigonometry. In the module The calculus of trigonometric functions, they are used to find, among other things, the derivative of the sine function.
To prove the cos(A−B) formula, from which we can obtain the other expansions, we return to the circle definition of the trigonometric functions.
Consider two points P(cosA,sinA) and Q(cosB,sinB) on the unit circle, making angles A and B respectively with the positive x-axis.
Detailed description of diagram
We will calculate the distance PQ in two ways and then equate the results. First we apply the cosine rule to the triangle OPQ. Note that, in the diagram above, ∠POQ=A−B. In general, it is always the case that cos(∠POQ)=cos(A−B). So the cosine rule gives
PQ2=12+12−2×1×1×cos(A−B)=2−2cos(A−B).On the other hand, using the square of the distance formula from coordinate geometry,
PQ2=(cosB−cosA)2+(sinB−sinA)2=cos2A+sin2A+cos2B+sin2B−2cosAcosB−2sinAsinB=2−2(cosAcosB+sinAsinB).Equating the two expressions for PQ2, we have
cos(A−B)=cosAcosB+sinAsinB.We can easily obtain the formula for cos(A+B) by replacing B with −B in the formula for cos(A−B) and recalling that cos(−θ)=cosθ (the cosine function is an even function) and sin(−θ)=−sinθ (the sine function is an odd function).
Hence
cos(A+B)=cosAcos(−B)+sinAsin(−B)=cosAcosB−sinAsinB.Using the identity sinθ=cos(90∘−θ), we can show that
sin(A+B)=sinAcosB+cosAsinB,sin(A−B)=sinAcosB−cosAsinB.These four compound angle formulas are important and the student should remember them. Most other trigonometric identities can be derived from these and the standard Pythagorean identity cos2θ+sin2θ=1.
Exercise 10
Use the identity sinθ=cos(90∘−θ) to derive the sine expansions.
The following exercise gives a simple geometric derivation of the sine expansion.
Exercise 11
Fix acute angles α and β. Construct a triangle ABC as shown in the following diagram. (Start by drawing the line interval CD. Then construct the right-angled triangles BCD and ACD.)
- Prove that y=acosα and y=bcosβ.
- By comparing areas, show that 12absin(α+β)=12aysinα+12bysinβ.
- Deduce the expansion formula for sin(α+β).
Using the compound angle formulas, we can extend the range of angles for which we can obtain exact values for the trigonometric functions.
Example
Find the exact value of
- cos75∘
- sin75∘
- cos105∘.
solution
-
cos75∘=cos(45∘+30∘)=cos45∘cos30∘−sin45∘sin30∘=1√2√32−1√212=14(√6−√2)
-
sin75∘=sin(45∘+30∘)=sin45∘cos30∘+cos45∘sin30∘=1√2√32+1√212=14(√6+√2)
- cos105∘=cos(45∘+60∘)=cos45∘cos60∘−sin45∘sin60∘=1√212−1√2√32=14(√2−√6)
(Note that we could obtain cos105∘ directly from cos75∘, since the two angles are supplementary.)
We can also find expansions for tan(A+B) and tan(A−B). Recalling that tanθ=sinθcosθ, we can write
tan(A+B)=sin(A+B)cos(A+B)=sinAcosB+cosAsinBcosAcosB−sinAsinB.Dividing the numerator and denominator by cosAcosB, we obtain
tan(A+B)=sinAcosA+sinBcosB1−sinAsinBcosAcosB=tanA+tanB1−tanAtanB.Since tan(−B)=−tanB, we have
tan(A−B)=tanA−tanB1+tanAtanB.Note carefully the pattern with the signs.
Exercise 12
Find the exact value of tan15∘.
Putting A=B=θ in the expansion formula for tan(A+B), we obtain
tan2θ=2tanθ1−tan2θ.Exercise 13
Show that t=tan6712∘ satisfies the quadratic equation t2−2t−1=0 and hence find its exact value.
The angle between two lines
The tangent expansion formula can be used to find the angle, or rather the tangent of the angle, between two lines.
Detailed description of diagram
Suppose two lines ℓ and n with gradients m1 and m2, respectively, meet at the point P. The gradient of a line is the tangent of the angle it makes with the positive x-axis. So, if ℓ and n make angles α and β, respectively, with the positive x-axis, then tanα=m1 and tanβ=m2. We will assume for the moment that α>β, as in the diagram above.
Now, if γ is the angle between the lines (as shown), then γ=α−β. Hence
tanγ=tan(α−β)=tanα−tanβ1+tanαtanβ=m1−m21+m1m2, provided m1m2≠−1. If m1m2=−1, the two lines are perpendicular and tanγ is not defined.In general, the above formula may give us a negative number, since it may be the tangent of the obtuse angle between the two lines.
Hence, if we are interested only in the acute angle, since tan(180∘−θ)=−tanθ, we can take the absolute value and say that, if γ is the acute angle between the two lines, then
tanγ=|m1−m21+m1m2|,provided the lines are not perpendicular.
Example
Find, to the nearest degree, the acute angle between the lines y=2x−1 and y=3x+4.
Solution
Here m1=2 and m2=3. So, if θ is the acute angle between the two lines, we have
tanθ=|2−31+6|=17 and therefore θ≈8∘.Exercise 14
Find the two values of m such that the angle between the lines y=mx and y=2x is 45∘. What is the relationship between the two lines you obtain?
If we define the angle between two curves at a point of intersection to be the angle between their tangents at that point, then the above formula — along with some differential calculus — can be used to find that angle.