Loading [MathJax]/jax/output/HTML-CSS/jax.js

Content

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.

Right –angled triangle ABC.
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. (1sinθ)(1+sinθ)=cos2θ
  2. 2cos3θcosθsinθcos2θsin3θ=cotθ.

Solution

  1. LHS=(1sinθ)(1+sinθ)=1sin2θ(difference of two squares)=cos2θ(Pythagorean identity)=RHS
  2. LHS=2cos3θcosθsinθcos2θsin3θ=cosθ(2cos2θ1)sinθ(cos2θsin2θ)=cosθ(2cos2θ1)sinθ(cos2θ(1cos2θ))=cosθ(2cos2θ1)sinθ(2cos2θ1)=cosθsinθ=cotθ=RHS

Exercise 7

Prove that

1secθ+tanθ=cosθ1+sinθ.

Screencast of exercise 7 mp4 of screencast of exercise 7

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.

Triangle ABC, isosceles triangle with AB = AC = 1. D is the midpoint of BC with AD perpendicular to BC. Angle BAD = angle CAD = theta.

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+124sin2θ2×1×1=24sin2θ2=12sin2θ.

Replacing 1 by cos2θ+sin2θ, we arrive at the double angle formula

cos2θ=cos2θsin2θ.

This may also be written as cos2θ=2cos2θ1 or cos2θ=12sin2θ, 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=2cos2θ1 and writing x for cos2212, we obtain

cos45=2x2112=2x212x2=1+22x=1+222, which simplifies to 122+2.

We can also find a double angle formula for sine using the same diagram.

Triangle ABC, isosceles triangle with AB = AC = 1. D is the midpoint of BC with AD perpendicular to BC. Angle BAD = angle CAD = theta.

In this case, we write down formulas for the area of ABC in two ways:

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=1cos2θsin2θ=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(1cos2θ). 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=2tanθ1tan2θ,for tanθ±1.
Summary of double angle formulas
sin2θ=2sinθcosθ
cos2θ=cos2θsin2θ=2cos2θ1=12sin2θ
tan2θ=2tanθ1tan2θ, 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(AB), cos(A+B) and cos(AB), 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(AB) 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.

Circle with radius of 1, centre the origin O.
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=AB. In general, it is always the case that cos(POQ)=cos(AB). So the cosine rule gives

PQ2=12+122×1×1×cos(AB)=22cos(AB).

On the other hand, using the square of the distance formula from coordinate geometry,

PQ2=(cosBcosA)2+(sinBsinA)2=cos2A+sin2A+cos2B+sin2B2cosAcosB2sinAsinB=22(cosAcosB+sinAsinB).

Equating the two expressions for PQ2, we have

cos(AB)=cosAcosB+sinAsinB.

We can easily obtain the formula for cos(A+B) by replacing B with B in the formula for cos(AB) 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)=cosAcosBsinAsinB.

Using the identity sinθ=cos(90θ), we can show that

sin(A+B)=sinAcosB+cosAsinB,sin(AB)=sinAcosBcosAsinB.

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(90θ) 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.)

Triangle ABC, AC = b, BC = a. D is a point on AB such that CD is perpendicular to AB. CD = y. Angle BCD = alpha and angle ACD = beta.

  1. Prove that y=acosα and y=bcosβ.
  2. By comparing areas, show that 12absin(α+β)=12aysinα+12bysinβ.
  3. 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

  1. cos75
  2. sin75
  3. cos105.

solution

  1. cos75=cos(45+30)=cos45cos30sin45sin30=12321212=14(62)
  2. sin75=sin(45+30)=sin45cos30+cos45sin30=1232+1212=14(6+2)
  3. cos105=cos(45+60)=cos45cos60sin45sin60=12121232=14(26)

(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(AB). Recalling that tanθ=sinθcosθ, we can write

tan(A+B)=sin(A+B)cos(A+B)=sinAcosB+cosAsinBcosAcosBsinAsinB.

Dividing the numerator and denominator by cosAcosB, we obtain

tan(A+B)=sinAcosA+sinBcosB1sinAsinBcosAcosB=tanA+tanB1tanAtanB.

Since tan(B)=tanB, we have

tan(AB)=tanAtanB1+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=2tanθ1tan2θ.

Exercise 13

Show that t=tan6712 satisfies the quadratic equation t22t1=0 and hence find its exact value.

Screencast of exercise 13 mp4 of screencast of exercise 13

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.

Line marked n, intercepting negative x axis and positive y axis angle created by line and x axis marked as beta.
Detailed description of diagram

Suppose two lines 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 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=αβ. Hence

tanγ=tan(αβ)=tanαtanβ1+tanαtanβ=m1m21+m1m2, provided m1m21. 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(180θ)=tanθ, we can take the absolute value and say that, if γ is the acute angle between the two lines, then

tanγ=|m1m21+m1m2|,

provided the lines are not perpendicular.

Example

Find, to the nearest degree, the acute angle between the lines y=2x1 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=|231+6|=17 and therefore θ8.

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.

Next page - Content - Radian measure