[Fig. 1.19]
Trigonometry begins with a right triangle. The size of the triangle is not as important as the angles. We focus on one particular angle—call it θ—and on the ratios between the three sides x, y, r. The ratios don't change if the triangle is scaled to another size. Three sides give six ratios, which are the basic functions of trigonometry:
[Fig. 1.19]
Ofcourse those six ratios are not independent. The three on the right come directly from the three on the left. And the tangent is the sine divided by the cosine:
tan θ = sin θ ⁄ cos θ = (y/r) ⁄ (x/r) = y ⁄ x
Note that "tangent of an angle" and "tangent to a circle" and "tangent line to a graph" are different uses of the same word. As the cosine of θ goes to zero, the tangent of θ goes to infinity. The side x becomes zero, θ approaches 90°, and the triangle is infinitely steep. The sine of 90° is y/r = 1.
Triangles have a serious limitation. They are excellent for angles up to 90°, and they are OK up to 180°, but after that they fail. We cannot put a 240° angle into a triangle. Therefore we change now to a circle.
[Fig. 1.20]
Angles are measured from the positive x axis (counterclockwise). Thus 90° is straight up, 180° is to the left, and 360° is in the same direction as 0°. (Then 450° is the same as 90°.) Each angle yields a point on the circle of radius r. The coordinates x and y of that point can be negative (but never r). As the point goes around the circle, the six ratios cos θ, sin θ, tan θ, .. . trace out six graphs. The cosine waveform is the same as the sine waveform—just shifted by 90°.
One more change comes with the move to a circle. Degrees are out. Radians are in. The distance around the whole circle is 2πr. The distance around to other points is θr. We measure the angle by that multiple θ. For a half-circle the distance is πr, so the angle is π radians–which is 180°. A quarter-circle is π/2 radians or 90°. The distance around to angle θ is r times θ.
When r = 1 this is the ultimate in simplicity: The distance is θ. A 45° angle is 1/8 of a circle and 2π/8 radians—and the length of the circular arc is 2π/8. Similarly for 1°:
360° = 2π radians 1° = 2π/360 radians 1 radian = 360/2π degrees.
An angle going clockwise is negative. The angle -π/3 is -60° and takes us 1/6 of the wrong way around the circle. What is the effect on the six functions?
Certainly the radius r is not changed when we go to -θ. Also x is not changed (see Figure 1.20a). But y reverses sign, because -θ is below the axis when +θ is above. This change in y affects y/r and y/x but not x/r:
cos(-θ) = cos θ sin(-θ) = -sin θ tan(-θ) = -tan θ.
The cosine is even (no change). The sine and tangent are odd (change sign).
The same point is 5/6 of the right way around. Therefore 5/6 of 2π radians (or 300°) gives the same direction as -π/3 radians or -60°. A difference of 2π makes no difference to x, y, r. Thus sin θ and cos θ and the other four functions have period 2π. We can go five times or a hundred times around the circle, adding 10π or 200π to the angle, and the six functions repeat themselves.
EXAMPLE Evaluate the six trigonometric functions at θ = 2π/3 (or θ = -4π/3). This angle is shown in Figure 1.20a (where r = 1). The ratios are
cos θ = x/r = -1/2 sin θ = y/r = √3/2 tan θ = y/x = -√3
sec θ = -2 csc θ = 2/√3 cot θ = -1/√3
Those numbers illustrate basic facts about the sizes of four functions:
|cos θ| ≤ 1 |sin θ| ≤ 1 |sec θ| ≥ 1 |csc θ| ≥ 1
The tangent and cotangent can fall anywhere, as long as cot θ = 1/tan θ.
The numbers reveal more. The tangent -√3 is the ratio of sine to cosine. The secant -2 is 1/cos θ. Their squares are 3 and 4 (differing by 1). That may not seem remarkable, but it is. There are three relationships in the squares of those six numbers, and they are the key identities of trigonometry:
cos² θ + sin² θ = 1 1 + tan² θ = sec² θ cot² θ + 1 = csc² θ
Everything flows from the Pythagoras formula x² + y² = r². Dividing by r² gives (x/r)² + (y/r)² = 1. That is cos² θ + sin² θ = 1. Dividing by x² gives the second identity, which is 1 + (y/x)² = (r/x)². Dividing by y² gives the third. All three will be needed throughout the book—and the first one has to be unforgettable.
To compute the distance between points we stay with Pythagoras. The points are in Figure 1.21a. They are known by their x and y coordinates, and d is the distance between them. The third point completes a right triangle.
For the x distance along the bottom we don't need help. It is x2−x1 (or |x2−x1| since distances can't be negative). The distance up the side is ly2−y1|. Pythagoras immediately gives the distance d:
distance between points = d = √{(x2−x1)²+ (y2−y1)²}. (1)
[Fig. 1.21]
By applying this distance formula in two identical circles, we discover the cosine of s−t. (Subtracting angles is important.) In Figure 1.21b, the distance squared is
d² = (change in x)² + (change in y)² = (cos s − cos t)² + (sin s − sin t)². (2)
Figure 1.21c shows the same circle and triangle (but rotated). The same distance squared is
d²= (cos(s−t)−1)² + (sin(s−t))². (3)
Now multiply out the squares in equations (2) and (3). Whenever (cosine)² + (sine)² appears, replace it by 1. The distances are the same, so (2) = (3):
(2) = 1 + 1 − 2 cos s cos t − 2 sin s sin t
(3) = 1 + 1 − 2cos(s−t)
After canceling 1 + 1 and then -2, we have the "addition formula" for cos(s−t):
The cosine of s−t equals cos s cos t + sin s sin t. (4)
The cosine of s+t equals cos s cos t − sin s sin t. (5)
The easiest is t = 0. Then cos t = 1 and sin t = 0. The equations reduce to cos s = cos s.
To go from (4) to (5) in all cases, replace t by -t. No change in cos t, but a "minus" appears with the sine. In the special case s = t, we have cos(t + t)= (cos t)(cos t) − (sin t)(sin t). This is a much-used formula for cos 2t:
Double angle: cos 2t = cos² t − sin² t = 2 cos² t − 1 = 1 − 2 sin² t. (6)
I am constantly using cos² t + sin² t = 1, to switch between sines and cosines.
We also need addition formulas and double-angle formulas for the sine of s−t and s + t and 2t. For that we connect sine to cosine, rather than (sine)² to (cosine)². The connection goes back to the ratio y/r in our original triangle. This is the sine of the angle θ and also the cosine of the complementary angle π/2 − θ:
sin θ = cos(π/2−θ) and cos θ = sin(π/2−θ). (7)
The complementary angle is π/2−θ because the two angles add to π/2 (a right angle). By making this connection in Problem 19, formulas (4-5-6) move from cosines to sines:
sin(s−t) = sin s cos t − cos s sin t (8)
sin(s+t) = sin s cos t + cos s sin t (9)
sin 2t = sin(t + t) = 2 sin t cos t (10)
I want to stop with these ten formulas, even if more are possible. Trigonometry is full of identities that connect its six functions—basically because all those functions come from a single right triangle. The x, y, r ratios and the equation x² + y² = r² can be rewritten in many ways. But you have now seen the formulas that are needed by calculus.* They give derivatives in Chapter 2 and integrals in Chapter 5. And it is typical of our subject to add something of its own—a limit in which an angle approaches zero. The essence of calculus is in that limit.
Review of the ten formulas Figure 1.22 shows d² = (0−½)² + (1−√3/2)².
| cos π/6 = cos π/2 cos π/3 + sin π/2 sin π/3 | (s − t) | sin π/6 = sin π/2 cos π/3 − cos π/2 sin π/3 |
| cos 5π/6 = cos π/2 cos π/3 − sin π/2 sin π/3 | (s + t) | sin 5π/6 = sin π/2 cos π/3 + cos π/2 sin π/3 |
| cos 2(π/3) = cos² π/3 − sin² π/3 | (2t) | sin 2(π/3) = 2 sin² π/3 cos² π/3 |
| cos π/6 = sin π/3 = √3/2 | (π/2 − θ) | sin π/6 = cos π/3 = 1/2 |
[Fig. 1.22]
