Difficulty: Intermediate
Prerequisites: Section 1.3 (instantaneous velocity). Some prior exposure to sine, cosine, and basic right-triangle trigonometry is helpful but not strictly required.
Section 1.4 introduces circular motion as a way to understand sine and cosine functions. A ball travelling counterclockwise around the unit circle at speed 1 has position (cos t, sin t). From this single picture you can read off the derivatives of sin t and cos t before any formal rules are developed. Section 1.5 collects the essential trigonometric identities and formulas that calculus uses repeatedly. Together, these two sections give you everything you need about trig for the rest of the course.
A ball on the unit circle at angle t has coordinates x = cos t, y = sin t. Its velocity is tangent to the circle: horizontal velocity is -sin t, vertical velocity is cos t. This means the derivative of sin t is cos t and the derivative of cos t is -sin t. Section 1.5 then reviews the key trig identities: Pythagorean, double-angle, and addition formulas.
Radian
The angle measure where the arc length on a unit circle equals the angle. A full circle is 2pi radians (360 degrees). Calculus always uses radians because the derivative formulas only work cleanly in radians.
Unit circle
A circle of radius 1, centred at the origin. The coordinates of a point at angle t are (cos t, sin t).
sin t (sine)
The y-coordinate of a point at angle t on the unit circle. It oscillates between -1 and 1 with period 2pi.
cos t (cosine)
The x-coordinate of a point at angle t on the unit circle. Same oscillation as sine, shifted by pi/2.
Period
The smallest time interval after which a function repeats. For sin t and cos t, the period is 2pi. For sin kt, the period is 2pi/k.
Simple harmonic motion
Up-and-down motion described by f(t) = sin t (or cos t). The "shadow" of circular motion projected onto a vertical (or horizontal) line.
Tangent, secant, cosecant, cotangent
tan t = sin t / cos t, sec t = 1 / cos t, csc t = 1 / sin t, cot t = cos t / sin t. These are ratios built from sine and cosine. In right-triangle terms: tangent = opposite/adjacent, secant = hypotenuse/adjacent, and so on.
Amplitude
The maximum displacement from the centre line of an oscillation. For A sin(kt), the amplitude is A.
A ball starts at (1, 0) and moves counterclockwise at speed 1 (1 metre per second on a unit circle).
At time t, the ball has swept through an angle of t radians.
Position: x = cos t, y = sin t.
One full revolution takes 2pi seconds (the circumference of the unit circle is 2pi).
The velocity vector is tangent to the circle, perpendicular to the radius.
Horizontal velocity: dx/dt = -sin t. Vertical velocity: dy/dt = cos t.
This immediately gives the derivatives: the derivative of sin t is cos t, and the derivative of cos t is -sin t.
At t = 0 the ball is at (1, 0), moving straight up: horizontal velocity = 0, vertical velocity = 1. This matches -sin 0 = 0 and cos 0 = 1.
Speed k (radians per second): position is (cos kt, sin kt). The angle at time t is kt radians.
Velocity: horizontal = -k sin kt, vertical = k cos kt. The speed is k (as expected).
Radius R: position is (R cos kt, R sin kt). Speed is kR.
Velocity: horizontal = -Rk sin kt, vertical = Rk cos kt.
If you watch only the y-coordinate of the ball, you see f(t) = sin t: simple harmonic motion.
Its velocity is v(t) = cos t.
A speeded-up version is f(t) = sin 2t, with velocity v(t) = 2 cos 2t.
sin t = opposite / hypotenuse, cos t = adjacent / hypotenuse, tan t = opposite / adjacent.
The reciprocals: csc t = 1/sin t, sec t = 1/cos t, cot t = 1/tan t.
On the unit circle these extend to all angles, not just acute ones.
sin^2(t) + cos^2(t) = 1. This comes directly from x^2 + y^2 = 1 on the unit circle.
Dividing by cos^2(t): 1 + tan^2(t) = sec^2(t).
Dividing by sin^2(t): 1 + cot^2(t) = csc^2(t).
cos(s - t) = cos s cos t + sin s sin t.
cos(s + t) = cos s cos t - sin s sin t.
sin(s + t) = sin s cos t + cos s sin t.
sin(s - t) = sin s cos t - cos s sin t.
sin 2t = 2 sin t cos t.
cos 2t = cos^2(t) - sin^2(t) = 2 cos^2(t) - 1 = 1 - 2 sin^2(t).
From cos 2t = 2 cos^2(t) - 1, rearrange to get the half-angle formula: cos^2(t) = (1 + cos 2t)/2. This formula is essential for integration.
Distance from (x_1, y_1) to (x_2, y_2) = sqrt[(x_2 - x_1)^2 + (y_2 - y_1)^2].
Example: distance from (2, 5) to (3, 3) = sqrt(1 + 4) = sqrt(5).
x = R\cos(kt), \quad y = R\sin(kt)Position of a ball on a circle of radius R moving at k radians per second.
\frac{dx}{dt} = -Rk\sin(kt), \quad \frac{dy}{dt} = Rk\cos(kt)Velocity components. Speed = kR.
\sin^2 t + \cos^2 t = 1The Pythagorean identity. Memorise this.
\cos 2t = 2\cos^2 t - 1 = 1 - 2\sin^2 tDouble-angle formula for cosine. Rearranges into the half-angle formulas needed for integration.
\cos^2 t = \frac{1 + \cos 2t}{2}Half-angle formula. You will use this when integrating cos^2(t).
Circular motion describes anything that rotates: wheels, gears, turbines, satellites, and the hands of a clock. Harmonic motion (the sine/cosine oscillation) describes springs, pendulums, sound waves, and AC electricity. The derivative rules for sin and cos are what make all of that analysis possible.
Students often forget that calculus requires radians, not degrees. If you use degrees, the derivative of sin t is not cos t; it picks up a factor of pi/180, which ruins every formula.
The derivative of cos t is -sin t, not +sin t. The minus sign is easy to drop and frequently tested.
sin^2(t) + cos^2(t) = 1 is not an equation to solve; it is always true for every value of t. Students sometimes try to "solve" it for t, which is meaningless.
Period and frequency are inverses. The period of sin(kt) is 2pi/k, not 2pi times k.
⚠️ The derivatives of sin t and cos t (and the sign in the cos derivative) appear on virtually every calculus exam.
⚠️ The identity cos^2(t) = (1 + cos 2t)/2 is needed for integration in Chapter 5. Learn it now.
⚠️ Expect questions that combine circular motion with the velocity ideas from 1.1 and 1.3: given the angle, find position and velocity components.
Fill in the blank: One revolution around the unit circle at speed 1 takes ______ seconds. 2pi.
True or False: The derivative of sin t is -cos t. False. It is +cos t.
Fill in the blank: The position of a ball on the unit circle at time t = pi is (______, ______). (-1, 0).
True or False: sin^2(t) + cos^2(t) = 1 only when t = 0. False. It is true for all t.
Fill in the blank: cos^2(t) = (1 + ______)/2. cos 2t.
Q: A ball moves counterclockwise around the unit circle at speed 4 m/s, starting at angle 0. What are its coordinates at time t?
A: (cos 4t, sin 4t). The angle at time t is 4t radians.
Q: What are the horizontal and vertical velocities of the ball in the previous question?
A: Horizontal: -4 sin 4t. Vertical: 4 cos 4t.
Q: Simplify sin(theta + 3pi/2).
A: sin theta cos(3pi/2) + cos theta sin(3pi/2) = (sin theta)(0) + (cos theta)(-1) = -cos theta.
Q: Find the distance from (1, 0) to (0, 1) along a straight line.
A: sqrt(1^2 + 1^2) = sqrt(2).
Q: Express sin 3t in terms of sin t and cos t.
A: sin 3t = sin(2t + t) = sin 2t cos t + cos 2t sin t = 2 sin t cos^2(t) + (cos^2(t) - sin^2(t)) sin t = 3 sin t - 4 sin^3(t).
The derivatives of sin t and cos t established here are used throughout the course. Section 2.4 rederives them formally using limits. The trig identities, particularly cos^2(t) = (1 + cos 2t)/2, become essential integration tools in Chapter 5. The addition formulas are needed for solving differential equations in later courses.
Circular motion also connects to parametric equations (Chapter 12 in many texts): writing x = cos t, y = sin t is a parametric description of the circle.
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