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From Förberedande kurs i matematik 1

Theory

Exercises

Introduction

There are a variety of trigonometric formulas which relate the sine, cosine or tangent for an angle or multiples of an angle. They are usually known as the trigonometric identities. Here we will give some of these trigonometric relationships, and show how to derive them. There are many more than we can deal with in this course, but most can be derived from the so-called Pythagorean identity and the addition and subtraction formulas (see below), which are important to know by heart.

The Pythagorean identity

This identity is the most basic, but is in fact nothing more than Pythagorean theorem, applied to the unit circle. The right-angled triangle on the right shows that (sinv)2+(cosv)2=1, which is usually written as sin2v+cos2v=1.

Symmetries

With the help of the unit circle and by exploiting the symmetries we obtain a large number of relationships between the cosine and sine functions:

Instead of trying to learn all of these relationships by heart, it is better to learn how to derive them from the unit circle.

Reflection in the x-axis

When an angle v is reflected in the x-axis it becomes −v. Reflection does not affect the x- coordinate while the y-coordinate changes sign, so that cos(−v)sin(−v)=cosv,=−sinv.

Reflection in the y-axis

Reflection in the y-axis changes the angle from v to −v (the reflection makes an angle v with the negative x-axis). This reflection does not affect the y-coordinate, while the x-coordinate changes sign, so we see that cos(−v)sin(−v)=−cosv,=sinv.

Reflection in the line y = x

The angle v is changed to 2−v (the reflected line makes an angle v with the positive y-axis). This reflection swaps the x- and y-coordinates, so that cos2−vsin2−v=sinv.=cosv.

Rotation by an angle of 2

A rotation 2 of the angle v means that the angle becomes v+2. The rotation turns the x-coordinate into the new y-coordinate and the y-coordinate turns into the new x-coordinate with the opposite sign, so that cosv+2sinv+2=−sinv,=cosv.

Alternatively, we can derive these relationships by reflecting and/or shifting the graph of sine and cosine. For instance, if we want to express cosv as the sine of an angle, we can shift the graph of sinv along in the v direction, so that it coincides with the graph of cosv. This can be done in several ways, but the most natural is to write cosv=sin(v+2). To avoid mistakes, we can check that this is true for several different values of v.

Check:  cos0=sin(0+2)=1.

The addition, subtraction and double-angle formulas

We often need to deal with expressions in which two or more angles are involved, such as sin(u+v). We will then need the so-called "addition formulas" . For sine and cosine the formulas are:

If we want to know the sine or cosine of a double angle, that is sin2v or cos2v, we can use the addition formulas above to get the double-angle formulas:

From these relationships, we can also derive formulas for half angles. By replacing 2v by v, and consequently v by v2, in the formula for cos2v we get that

cosv=cos2v2 – sin2v2.

If we want a formula for sin(v2) we use the Pythagorean identity to write cos2(v2) in terms of sin2(v2). Then

cosv=1–sin2v2–sin2v2=1–2sin2v2

so that

Similarly, we can write sin2(v2) in terms of cos2(v2). Then we will arrive at

Exercises