A. Rate of Change · Derivatives of Exponential and Sinusoidal FunctionsInvestigating the Derivative of sin x
A2.4About 70 minutes
Learning goals- I can build a table of instantaneous rates of change of y = sin x and graph them.
- I can recognize that the graph of the derivative of sin x is the graph of cos x.
- I can explain, using the limits of (sin h)/h and (cos h − 1)/h, why d/dx sin x = cos x when x is in radians.
- I can differentiate functions involving sin x and find tangent lines to y = sin x.
Words to know
radian
The angle that cuts off an arc equal to the radius. 2π radians = 360°. Calculus rules for sine and cosine need radians.
sinusoidal function
A function whose graph is a sine or cosine wave, such as y = sin x or y = 3 cos(2x) + 1.
derivative of sin x
d/dx sin x = cos x (x in radians).
compound angle formula
sin(a + b) = sin a cos b + cos a sin b, from MHF4U.
limit
The value a quantity approaches as the input approaches some value, such as (sin h)/h → 1 as h → 0.
Waves have slopes too
A child on a playground swing, a buoy bobbing on Lake Ontario, the voltage in a household outlet: all follow sine waves. How fast are they changing at each moment? To answer, we need the derivative of sin x.
Remember from MHF4U: in Radian Measure and Trigonometric Functions you graphed y = sin x in radians, with period 2π. In the Rates of Change unit you estimated that a sine curve changes fastest where it crosses its midline and has rate 0 at its peaks and troughs. Today you will turn that into an exact rule.
Set your calculator to radian mode. Everything in this lesson uses radians.
A table of tangent slopes
Expectation A2.4 asks you to find the derivative by investigation. For each x, estimate the slope with the difference quotient [sin(x + h) − sin x] ÷ h, using h = 0.0001.
| x | sin x | slope (h = 0.0001) | cos x |
|---|
| 0 | 0 | 1.0000 | 1 |
| π/6 ≈ 0.524 | 0.5 | 0.8660 | 0.8660 |
| π/3 ≈ 1.047 | 0.8660 | 0.5000 | 0.5 |
| π/2 ≈ 1.571 | 1 | 0.0000 | 0 |
| 2π/3 ≈ 2.094 | 0.8660 | −0.5000 | −0.5 |
| 5π/6 ≈ 2.618 | 0.5 | −0.8661 | −0.8660 |
| π ≈ 3.142 | 0 | −1.0000 | −1 |
| 3π/2 ≈ 4.712 | −1 | 0.0000 | 0 |
| 2π ≈ 6.283 | 0 | 1.0000 | 1 |
The derivative of sine
d/dx sin x = cos x (x in radians).
On a graphing tool you can check it by graphing y = sin x, asking for its derivative (or plotting the slope of a sliding tangent), and seeing the graph of y = cos x appear.
Why it works: the algebra
Use the definition of the derivative and the compound angle formula from MHF4U Trigonometric Identities and Equations, sin(x + h) = sin x cos h + cos x sin h:
[sin(x + h) − sin x] ÷ h = [sin x cos h + cos x sin h − sin x] ÷ h
= sin x × (cos h − 1)/h + cos x × (sin h)/h.
As h → 0, only two limits matter. They do not depend on x, and a calculator in radian mode shows them clearly:
| h | (sin h)/h | (cos h − 1)/h |
|---|
| 0.5 | 0.958851 | −0.244835 |
| 0.1 | 0.998334 | −0.049958 |
| 0.01 | 0.999983 | −0.005000 |
| 0.001 | 1.000000 (rounded) | −0.000500 |
So the difference quotient approaches sin x × 0 + cos x × 1 = cos x.
Why radians? The fact (sin h)/h → 1 is true only when h is measured in radians. For a small angle, the arc length h and the height sin h on a unit circle are almost equal. In degrees, the slope of y = sin x at 30° is about 0.01511, not cos 30° ≈ 0.866: it is (π/180) cos 30°. Radians make the rule clean, which is why calculus always uses them.
Worked exampleExample 1: Basic derivatives
Differentiate (a) y = 3 sin x (b) y = sin x + x² (c) y = 5x − 2 sin x
- (a) Constant multiple rule: y′ = 3 cos x.
- (b) Sum rule: y′ = cos x + 2x.
- (c) y′ = 5 − 2 cos x.
(a) 3 cos x (b) cos x + 2x (c) 5 − 2 cos x
Worked exampleExample 2: A tangent line
Find the equation of the tangent to y = sin x at x = π/3.
- Point: (π/3, sin(π/3)) = (π/3, √3/2).
- Slope: y′ = cos x, so m = cos(π/3) = 1/2.
- y − √3/2 = (1/2)(x − π/3).
- y = (1/2)x − π/6 + √3/2 ≈ 0.5x + 0.342.
y = (1/2)x + √3/2 − π/6 (about y = 0.5x + 0.342).
Worked exampleExample 3: Where is the tangent horizontal?
Find all x in [0, 2π] where y = sin x has a horizontal tangent, and where its slope is 1/2.
- Horizontal: cos x = 0, so x = π/2 or x = 3π/2 (the peak and the trough).
- Slope 1/2: cos x = 1/2. From the special triangles and the unit circle (MHF4U), x = π/3 or x = 5π/3.
Horizontal at x = π/2 and 3π/2; slope 1/2 at x = π/3 and 5π/3.
Worked exampleExample 4: Mixing with earlier rules
For f(x) = 2 sin x − x, find f′(π) and say whether f is increasing or decreasing there.
- f′(x) = 2 cos x − 1.
- f′(π) = 2(−1) − 1 = −3.
f′(π) = −3, so f is decreasing at x = π.
Hands-on activityMeasure the slopes of a sine wave by hand
You need: graph paper, a ruler, a pencil, a calculator in radian mode, and a piece of string about 50 cm long with a small weight on the end (for example a key or a metal washer) for a pendulum.
Part A: tangents on paper
- Draw x-axis from 0 to 6.5 and y-axis from −1.2 to 1.2. Use the same scale on both axes (for example 2 cm per unit). This matters: slopes only read correctly with equal scales.
- Plot y = sin x at x = 0, 0.5, 1, 1.5, …, 6.5 and draw a smooth curve.
- At x = 0, 1, 2, 3, 4, 5 and 6, lay a ruler along the curve as a tangent and estimate its slope (rise ÷ run).
- Plot your slopes on the same grid. Then compute cos x at those x-values and compare.
Part B: pendulum
- Hold the string still and let the weight swing gently. Watch where it moves fastest and where it pauses.
What to look for: your slopes should be close to cos 0 = 1, cos 1 ≈ 0.54, cos 2 ≈ −0.42, cos 3 ≈ −0.99, cos 4 ≈ −0.65, cos 5 ≈ 0.28 and cos 6 ≈ 0.96. The pendulum pauses at the ends of its swing (largest displacement) and moves fastest through the middle (zero displacement), just as sin x has slope 0 at its peaks and slope ±1 at its zeros. A motion sensor in a classroom would show the velocity graph as a cosine-shaped wave.
Just for funWhy is the sine function never upset about its derivative? Because cos x is just sin x shifted left by π/2. It knows it's only a phase.
Your turn
1Differentiate y = 4 sin x.
2Differentiate y = x³ − sin x.
3Find the slope of y = sin x at x = π/4, exactly.
4Find the slope of y = sin x at x = 2 (radians), to three decimal places.
5Find the equation of the tangent to y = sin x at x = 0.
Point (0, 0), slope cos 0 = 1: y = x.
6Find all x in [0, 2π] where y = sin x has slope −1.
More practice: warm-up
2In what angle unit must x be measured for d/dx sin x = cos x to be true?
3What is the slope of y = sin x at its maximum point?
0 (the tangent is horizontal).
4Evaluate the limit of (sin h)/h as h → 0 (h in radians).
5Differentiate y = sin x + 7.
More practice: on your own
1Find the equation of the tangent to y = 2 sin x at x = π/6.
Point (π/6, 1); slope 2 cos(π/6) = √3. y = √3 x + 1 − √3π/6 ≈ 1.732x + 0.093.
2For f(x) = x + 2 sin x, find all x in [0, 2π] where the tangent is horizontal.
f′(x) = 1 + 2 cos x = 0 gives cos x = −1/2: x = 2π/3 and x = 4π/3.
3Explain using the derivative why y = x + sin x never decreases.
y′ = 1 + cos x. Since cos x ≥ −1, y′ ≥ 0 for every x, so the function never decreases (it has momentary flat spots where cos x = −1, at x = π, 3π, …).
4Use h = 0.001 to estimate the slope of y = sin x at x = 1, and compare with cos 1.
(sin 1.001 − sin 1)/0.001 ≈ 0.5399, and cos 1 ≈ 0.5403. They agree to about three decimal places.
5Find y″ for y = sin x, given that d/dx cos x = −sin x (next lesson).
y′ = cos x, so y″ = −sin x.
Challenge
1If x is in degrees, show that the slope of y = sin x at x = 30° is about 0.0151, and explain where the factor π/180 comes from.
x° = πx/180 radians, so sin(x°) = sin(πx/180). By the chain rule its derivative is (π/180) cos(πx/180). At x = 30: (π/180)(√3/2) ≈ 0.0151. The factor π/180 converts degrees to radians.
2Use the table of (sin h)/h to explain why sin(0.01) ≈ 0.01. How good is the approximation sin h ≈ h for h = 0.1?
(sin h)/h ≈ 1 for small h, so sin h ≈ h. For h = 0.1: sin 0.1 ≈ 0.09983, an error of about 0.00017 (0.17%). This 'small-angle approximation' is used in physics for pendulums.
3Find the points on y = sin x, 0 ≤ x ≤ 2π, where the tangent is parallel to the line y = x/2 + 3.
Slope 1/2: cos x = 1/2, so x = π/3 or 5π/3. Points (π/3, √3/2) and (5π/3, −√3/2).
Quiz
2. At which x-value is the slope of y = sin x equal to −1?
3. Why do calculus rules for sine use radians?
4. The tangent to y = sin x at x = 0 is:
5. If f(x) = 3 sin x + x², then f′(0) =
6. Where is y = sin x increasing most steeply on [0, 2π]?
Big idea (A2.4): the derivative of sin x is cos x. Students should see this three ways: from a table of slopes, from the graph (the slope curve of a sine wave is a cosine wave), and from the limit argument using (sin h)/h → 1. It only works in radians.
How to help: check that the calculator is in radian mode (a very common source of wrong answers). Ask your teen to point to the peaks of a sine graph and say what the derivative is there (0), and where the curve is steepest (at the zeros).
Common difficulties:
- Using degree mode: the slopes come out about 57 times too small.
- Mixing up the signs: d/dx sin x = cos x (positive), but d/dx cos x = −sin x (next lesson).
- Reading slopes from a graph whose axes have different scales.
- Forgetting exact values from the unit circle (MHF4U), such as cos(π/3) = 1/2.