Appendix B

Trigonometry.

Trigonometry looks at the relationships between the sides of right-triangles (triangles where one angle is \( 90^o \)).

Sin, Cos, Tan.

Look at the right-triangle in Figure 1. This has three angles \( a \), \( b \), and \( c \), and the lengths of the sides are \( A \) , \( O \), and \( H \). \( H \) is the longest side of the triangle, called the hypotenuse .

fig1 — **Figure 1** A right-triangle. The lengths of the three sides are \( A \), \( O \), and \( H \) . The angle between \( A \) and \( O \) is \( 90^o \).

The sine function of an angle \( a \) is defined as the length of the triangle's side opposite the angle, \( O \), divided by the length of the hypotenuse \( H \) :

\[\sin{a} = \frac{O}{H}\]

The cosine function of an angle \( a \) is defined as the length of the triangle's side adjacent to the angle, \( A \), divided by the length of the hypotenuse \( H \):

\[\cos{a} = \frac{A}{H}\]

The tangent function of an angle \( a \) is defined as the length of the triangle's side opposite the angle, \( O \), divided by the length of triangle's side adjacent to the angle, \( A \):

\[\tan{a} = \frac{O}{A}\]

If you double the size of the triangle, the angles are unchanged but the length of all sides are doubled. However, the ratios \( A/H \), \( O/H \), and \( O/A \) are unchanged, so these ratios are directly related to the angles.

The sine, cosine, and tangent of the angle \( b \) are

\[ \begin{align} \sin{b} &= A/H \\ \cos{b} &= O/H \\ \tan{b} &= A/O \\ \end{align} \]

Since \( b=90-a \), this shows that \( \sin{a}=O/H=\cos{b}=cos{(90-a)} \).

Remembering the trigonometric functions.

An easy way to remember the definitions of sine, cosine, and tangent is the phrase SOH CAH TOA. The SOH means the S ine is the Opposite side from the angle over the H ypotenuse; CAH means the Cosine is the Adjacent side to the angle over the Hypotenuse; and TOA means the Tangent is the Opposite side over the Adjacent side.

Degrees and Radians.

Typically, we use degrees to measure angles. There are \( 90 \) degrees in a right angle, \( 180 \) degrees in a half-circle, and \( 360 \) degrees in a full circle. (The choice of \( 360 \) may be due to there being about \( 360 \) days in a year, so a degree is about a day around the sun, or to the fact that ancient Babylonians liked multiples of \( 12 \), or both.)

Another measure of angle is the radian, based on arcs of a circle. In Figure 2, a slice of a circle with radius \( r \) is shown. The arc is the piece of the circle's circumference, and has a length of \( \alpha \). The angle \( a \) of this slice, measured in radians, is defined as

\[ a = \frac{\alpha}{r}\;\text{radians} \]

This definition of angle is sometimes mathematically useful. Since the circumference of a full circle is \( 2\pi r \), there are \( (2\pi r)/r=2\pi \) radians in a full circle, \( \pi \) radians in a half circles, and \( \pi/2 \) radians in a right angle. One radian is approximately \( 57 \) degrees.

Small-Angle Approximations.

Figure 3 shows a right-triangle with a very small angle \( a \), drawn inside an arc. When \( a \) is small, then the length of the adjacent side \( A \) and the length of the hypotenuse \( H \) are approximately the same; that is \( A\approx H \). In that case,

\[ \sin{a} = \frac{O}{H} \approx \frac{O}{A} = \tan{a} \]

fig3 — **Figure 3** A right-triangle with a piece of a circle superimposed. The hypotenuse \( H \) is equal to the radius of the circle \( r \). When the angle \( a \) is very small (narrow), then \( \alpha\approx O \) and \( A\approx r \). These approximations lead to the small angle approximations used in optics.

This is our first small-angle approximation:

\( \sin{a} \approx \tan{a} \) when \( a \) is small.

How small? Well, this approximation is very good when \( a<5^o \). For example, \( \sin{5^o}=0.0871557 \), and \( \tan{5^o}=0.0874887 \). The difference between the two is only \( 0.4\% \).

The size of the angle \( a \) in radians is \( a=\alpha/r \) radians; but the radius of the arc is also \( H \), so the angle is also \( a=\alpha/H \) radians. When the angle \( a \) is small, the arc length \( \alpha \) is approximately the same as the height of the triangle \( O \); that is \( \alpha\approx O \). In that case,

\[ \sin{a} = \frac{O}{H} \approx \frac{\alpha}{H} = \frac{\alpha}{r}= a \]

This is our second small-angle approximation:

\( \sin{a}\approx a \), in radians, when \( a \) is small.

Again, this approximation is very good for \( a<5^o \) .