Grade by Grade
B. Trigonometric Functions · Radian Measure and Trigonometric Functions

The Graphs of y = sin x and y = cos x in Radians

B2.1About 70 minutes
Learning goals
  • I can sketch y = sin x and y = cos x with radian measures, using five key points per cycle.
  • I can state the domain, range, period, amplitude, axis, zeros, maximum and minimum points and y-intercept of each function.
  • I can state the intervals where each function is increasing or decreasing.
  • I can use the graphs to estimate values and compare the two functions.

Words to know

periodic function
A function whose graph repeats at regular intervals.
period
The horizontal length of one cycle. For y = sin x and y = cos x it is 2π.
amplitude
Half the distance between the maximum and minimum values. For y = sin x and y = cos x it is 1.
equation of the axis
The horizontal line halfway between the maximum and minimum, here y = 0.
zero (of a function)
An x-value where the function equals 0; an x-intercept of the graph.
n ∈ ℤ
"n is an integer": n can be …, −2, −1, 0, 1, 2, … Used to describe infinitely many x-values at once.

From the unit circle to a wave

Picture a point travelling counterclockwise around the unit circle. After it has travelled an arc of length x, the angle is x radians and the point is at (cos x, sin x). Now plot the height of the point, sin x, against the distance travelled, x. You get the sine wave.

This is the big advantage of radians. On the x-axis, a distance of 1 really means 1 unit of arc, the same unit as on the y-axis. That is why calculus and physics always graph trigonometric functions in radians, and why x is a real number here, not just an angle.

Remember from Grade 11: in Sinusoidal Functions you graphed y = sin x and y = cos x with x in degrees, using key points every 90°. Now the key points are every π/2.

x0π/2π3π/22π
x (decimal)01.573.144.716.28
sin x010−10
cos x10−101
The five key points of one cycle. They come straight from the quadrantal points of the unit circle.
π/2π3π/22π1−1xy(0, 0)(π/2, 1)(π, 0)(3π/2, −1)(2π, 0)≈ 1.57≈ 3.14≈ 4.71≈ 6.28
One cycle of y = sin x from 0 to 2π. The five key points are a quarter period (π/2 ≈ 1.57) apart: zero, maximum, zero, minimum, zero.

Properties

Both functions repeat every 2π, because adding a full turn brings the point back to the same spot on the unit circle: sin(x + 2π) = sin x and cos(x + 2π) = cos x.

Propertyy = sin xy = cos x
Domain{x ∈ ℝ}{x ∈ ℝ}
Range{y ∈ ℝ | −1 ≤ y ≤ 1}{y ∈ ℝ | −1 ≤ y ≤ 1}
Period / amplitude / axis2π / 1 / y = 02π / 1 / y = 0
y-intercept01
Zerosx = nπ, n ∈ ℤx = π/2 + nπ, n ∈ ℤ
Maximum points(π/2 + 2nπ, 1)(2nπ, 1)
Minimum points(3π/2 + 2nπ, −1)(π + 2nπ, −1)
On 0 ≤ x ≤ 2π, increasing on[0, π/2] and [3π/2, 2π][π, 2π]
On 0 ≤ x ≤ 2π, decreasing on[π/2, 3π/2][0, π]
Key properties with radian measures. In each row, n ∈ ℤ means the pattern repeats in both directions.
−2π−ππ2π1−1xyy = sin x (solid)y = cos x (dashed)
y = sin x and y = cos x from −2π to 2π. The cosine graph is the sine graph moved π/2 to the left: every feature of cosine happens π/2 earlier.

Sketching in 30 seconds

To sketch either graph quickly: mark π/2, π, 3π/2 and 2π on the x-axis, draw dashed lines at y = 1 and y = −1, plot the five key points, and join them with a smooth, rounded curve (no sharp corners at the maximum and minimum). Then copy the pattern left and right.

Worked example

Example 1: Sketch over two cycles

Sketch y = sin x for −2π ≤ x ≤ 2π. List its zeros, maximum points and minimum points in this interval.

  1. Plot the key points for 0 to 2π: (0, 0), (π/2, 1), (π, 0), (3π/2, −1), (2π, 0).
  2. Subtract 2π from each x to get the cycle from −2π to 0: (−2π, 0), (−3π/2, 1), (−π, 0), (−π/2, −1), (0, 0).
  3. Join smoothly.

Zeros: −2π, −π, 0, π, 2π. Maximum points: (−3π/2, 1), (π/2, 1). Minimum points: (−π/2, −1), (3π/2, −1).

Worked example

Example 2: General zeros

Write an expression for all the zeros of y = cos x, and list the zeros between −π and 3π.

  1. Cosine is the x-coordinate on the unit circle; it is 0 at the top and the bottom, π/2 and 3π/2.
  2. These are π apart, so all zeros are x = π/2 + nπ, n ∈ ℤ.
  3. n = −1, 0, 1, 2 give x = −π/2, π/2, 3π/2, 5π/2. (n = 3 gives 7π/2 > 3π.)

x = π/2 + nπ, n ∈ ℤ; between −π and 3π: −π/2, π/2, 3π/2, 5π/2.

Worked example

Example 3: Reading the graph

Use the graph of y = sin x to find how many solutions sin x = 0.5 has for 0 ≤ x ≤ 2π. Then find them exactly.

  1. The horizontal line y = 0.5 crosses the first hump of the sine wave twice and misses the trough. So there are two solutions.
  2. From the unit circle, sin(π/6) = 1/2 and sin(5π/6) = 1/2.
  3. As decimals: π/6 ≈ 0.52 and 5π/6 ≈ 2.62. Both fit the picture: one on the way up, one on the way down.

Two solutions: x = π/6 and x = 5π/6. Solving equations like this is the focus of Trigonometric Identities and Equations.

Worked example

Example 4: Increasing and decreasing

On 0 ≤ x ≤ 2π, find where y = sin x and y = cos x are both decreasing.

  1. sin x decreases on [π/2, 3π/2].
  2. cos x decreases on [0, π].
  3. Both decrease where the intervals overlap: π/2 ≤ x ≤ π.

π/2 ≤ x ≤ π (quadrant II on the unit circle: the point moves left and down).

Just for fun

Sine and cosine are the same wave. Cosine just got up π/2 earlier, so it is always a quarter cycle ahead.

Hands-on activity

Unroll the unit circle

You need: a paper plate or a circle of radius 5 cm drawn on paper (trace a cup or use a compass), two sheets of paper taped end to end, a ruler, a pencil, a piece of string about 40 cm long, and a few strips of paper.

  1. Draw a horizontal axis about 35 cm long on the taped sheets. Draw a vertical axis at its left end.
  2. Around your circle, mark the 16 special angles (or the 12 multiples of π/6) as on your radian clock.
  3. Use the string to measure the arc from 0 to π/6 on the circle, and mark that same length along the x-axis. Keep going: mark π/3, π/2, and so on up to 2π. With r = 5 cm, 2π should land near 31.4 cm.
  4. For each point on the circle, measure its height above or below the horizontal diameter (positive above, negative below) and plot that height above the matching x-mark.
  5. Join the points smoothly. Repeat with the horizontal distance from the vertical diameter to draw the cosine graph.

What to look for: the x-axis distances are real arc lengths, which is exactly what a radian measures. The heights should rise to 5 cm (one radius) at π/2 and fall to −5 cm at 3π/2, giving a sine wave whose period is 2π radii.

Your turn

1

State the period, amplitude and range of y = cos x.

2

List the zeros of y = sin x for 0 ≤ x ≤ 4π.

3

For which x in [0, 4π] does y = sin x reach its maximum?

4

On 0 ≤ x ≤ 2π, where is y = cos x increasing?

5

State the y-intercept of y = cos x.

6

Estimate sin 2.5 from the graph, then check with a calculator.

7

Write all the minimum points of y = sin x using n ∈ ℤ.

More practice: warm-up

1

State the domain of y = sin x.

2

What is the maximum value of y = cos x?

3

Is (π, 0) on the graph of y = sin x?

4

Give the period of y = sin x as a decimal.

5

Is (π, −1) a maximum or a minimum point of y = cos x?

More practice: on your own

1

On −2π ≤ x ≤ 2π, list every x where sin x = 1.

2

On 0 ≤ x ≤ 2π, where are sin x and cos x both increasing?

3

Use the unit circle to explain why the range of y = sin x is −1 ≤ y ≤ 1.

4

How many solutions does sin x = 0.3 have for 0 ≤ x ≤ 4π? Explain with the graph.

5

Find the points of intersection of y = sin x and y = cos x for 0 ≤ x ≤ 2π.

Challenge

1

Use the graphs to explain why cos x = sin(x + π/2). Check it at x = 0, π/2 and π.

2

Show with the unit circle that sin(−x) = −sin x and cos(−x) = cos x. What does each fact say about the graphs?

3

For which x in [0, 2π] is sin x > cos x? Explain using your sketch.

Check your understanding

Quiz

1.

What is the period of y = sin x?

2.

Which expression gives all the zeros of y = cos x?

3.

For 0 < x < 2π, where does y = cos x have its minimum?

4.

On 0 ≤ x ≤ 2π, y = sin x is decreasing on...

5.

What is the range of y = sin x?

6.

On a decimal x-axis, the first maximum of y = sin x with x > 0 is at about...

For parents and teachers

Big idea (B2.1): y = sin x and y = cos x are waves with period 2π (about 6.28) and amplitude 1. The five key points of each cycle are π/2 apart. Cosine is sine shifted π/2 to the left.

How to help: Ask your teen to sketch both graphs from memory on scrap paper and mark π/2, π, 3π/2, 2π. Then ask a property ("Where are the zeros of cosine?") and have them point to it on the sketch.

Common difficulties:

  • Spacing: π is not 3 tick marks if each tick is π/2 (it is 2).
  • Drawing sharp, V-shaped peaks instead of rounded ones.
  • Mixing up the zeros of sine (nπ) and cosine (π/2 + nπ).
  • Writing increasing/decreasing intervals as y-values instead of x-values.