C. Trigonometric Functions · Sine FunctionsGraphing the Sine Function
C2.3 C2.4About 70 minutes
Learning goals- I can make a table of values for sin x from 0° to 360° and graph it.
- I can explain how the graph comes from heights on the unit circle.
- I can explain why f(x) = sin x is a function.
- I can use the graph and the 360° period to find sine values and angles.
Words to know
sine function
The function f(x) = sin x, which gives the sine of each angle x.
sine curve
The smooth wave-shaped graph of f(x) = sin x.
vertical line test
A graph shows a function if no vertical line meets it more than once.
degree mode
The calculator setting in which angles are measured in degrees (sin 90 = 1).
From a circle to a wave
In Lesson 3 you found that sin θ is the height of a point on a unit circle. Now imagine the point travelling around the circle while you record its height for every angle, then lay those heights out side by side along an x-axis. The result is one of the most famous graphs in mathematics: the sine curve. It describes tides, sound, light, alternating current and Ferris wheels.
Making a table of values
Use a calculator in degree mode to find sin x for angles from 0° to 360°. Round to three decimal places.
| x | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° |
|---|
| sin x | 0 | 0.5 | 0.707 | 0.866 | 1 | 0.866 | 0.707 | 0.5 | 0 |
| x | 180° | 210° | 225° | 240° | 270° | 300° | 315° | 330° | 360° |
|---|
| sin x | 0 | −0.5 | −0.707 | −0.866 | −1 | −0.866 | −0.707 | −0.5 | 0 |
Defining the sine function
The relationship between an angle x and its sine is written as the function
f(x) = sin x
where x is the angle in degrees.
Why is it a function? A relation is a function if each input has exactly one output. Every angle x has exactly one terminal arm, and one point where that arm meets the unit circle. That point has exactly one y-coordinate. So each x gives exactly one value of sin x. On the graph, every vertical line meets the curve at most once (the vertical line test).
Different angles can share the same sine (sin 30° = sin 150° = 0.5). That is allowed: a function may send several inputs to the same output. It just may not send one input to two outputs.
Worked exampleExample 1: Is it a function?
A student says, "sin x = 0.5 for x = 30° and x = 150°, so sin x is not a function." Explain the mistake.
- A function must give one output for each input.
- Here two different inputs (30° and 150°) give the same output (0.5). That does not break the rule.
- It would break the rule only if one angle, such as 30°, gave two different sine values. That never happens.
The student mixed up the direction. sin x is a function because each angle has only one sine; it is fine for two angles to share a sine.
Beyond 0° to 360°
Angles can be larger than 360° (more than one full turn) or negative (turning clockwise). The angle 390° = 360° + 30° has the same terminal arm as 30°, so sin 390° = sin 30° = 0.5. Similarly, −30° ends in the same place as 330°, so sin(−30°) = −0.5.
So the graph of f(x) = sin x repeats every 360° forever in both directions:
sin(x + 360°) = sin x for every x.
Worked exampleExample 2: Using the graph and the period
Use the graph or the table to find (a) sin 450°, (b) sin(−90°), (c) sin 600°, (d) every x from 0° to 360° where sin x = 0.866.
- (a) 450° = 360° + 90°, so sin 450° = sin 90° = 1.
- (b) −90° = 270° − 360°, so sin(−90°) = sin 270° = −1.
- (c) 600° = 360° + 240°, so sin 600° = sin 240° ≈ −0.866.
- (d) From the table, sin 60° ≈ 0.866 and sin 120° ≈ 0.866. On the graph, the horizontal line y = 0.866 crosses one cycle twice. So x = 60° and x = 120°.
(a) 1 (b) −1 (c) about −0.866 (d) 60° and 120°.
Worked exampleExample 3: Estimating from the graph
Use the graph of f(x) = sin x to estimate the values of x between 0° and 360° for which sin x = −0.3. Then check with a calculator.
- Draw (or imagine) the horizontal line y = −0.3. It is below the axis, so it crosses the curve between 180° and 360°: once a little after 180°, and once a little before 360°.
- Estimate from the graph: about 197° and 343°.
- Check: sin⁻¹(0.3) ≈ 17.5°, the related acute angle. Sine is negative in quadrants III and IV: 180° + 17.5° = 197.5° and 360° − 17.5° = 342.5°. Both estimates were close.
x ≈ 197.5° and x ≈ 342.5°.
Using technology
On a graphing calculator, set the mode to degrees, enter Y1 = sin(X), and use a window such as Xmin = 0, Xmax = 360, Xscl = 90, Ymin = −1.5, Ymax = 1.5. In a free online graphing tool, switch the angle setting to degrees. If the graph looks like a flat line or a jagged mess, check the angle mode and window.
Just for funWhy did the sine curve go to therapy?
It had too many ups and downs, but at least they were perfectly predictable.
Hands-on activityUnwrap a circle with string
You need: a round lid or plate, a piece of string or yarn, tape, a ruler, a sheet of paper and a pencil.
- Draw an x-axis across the paper. Mark 0° on the left and space marks every 30° up to 360° (for example 1.5 cm apart).
- Trace the lid on the left side of the paper, with its centre level with the x-axis. Draw the circle's horizontal diameter.
- Mark points on the circle every 30° (use a protractor, or fold the circle into twelfths).
- For each point, use the ruler to measure its height above (+) or below (−) the diameter, and plot that height above its angle on the x-axis.
- Join the points smoothly. Tape the string along your curve to see the wave.
What to look for: a smooth wave that starts at 0, reaches its highest point at 90°, crosses the axis at 180°, reaches its lowest point at 270° and returns to 0 at 360°. The highest height equals the radius of the lid.
Your turn
1Copy and complete: sin 0° = __, sin 90° = __, sin 180° = __, sin 270° = __, sin 360° = __.
2Explain in one or two sentences why f(x) = sin x is a function.
Each angle x has exactly one terminal arm and one point on the unit circle, so it has exactly one sine value. One input never gives two outputs.
3Find sin 420° without a calculator. Explain.
420° = 360° + 60°, so sin 420° = sin 60° ≈ 0.866.
4Find sin(−180°) and sin(−270°).
−180° ends at the same place as 180°: 0. −270° ends at the same place as 90°: 1.
5Use the table to find all x from 0° to 360° with sin x = −0.707.
sin 225° ≈ −0.707 and sin 315° ≈ −0.707: x = 225° and 315°.
6Between which two angles from 0° to 360° is sin x greater than 0.5?
sin x > 0.5 for 30° < x < 150°.
More practice: warm-up
1What is the largest value of sin x? At which angle from 0° to 360°?
2What is the smallest value of sin x? At which angle from 0° to 360°?
3How many degrees does it take for the sine graph to repeat?
4sin 30° = 0.5. What is sin 390°?
5Is sin 100° positive or negative?
Positive (quadrant II, above the axis).
More practice: on your own
1Find sin 750° and sin(−330°), showing how you used the period.
750° = 2(360°) + 30°, so sin 750° = sin 30° = 0.5. −330° = 30° − 360°, so sin(−330°) = sin 30° = 0.5.
2Estimate from the graph, then calculate, the values of x from 0° to 360° with sin x = 0.8 (nearest tenth).
Related angle sin⁻¹(0.8) ≈ 53.1°. Sine is positive in quadrants I and II: x ≈ 53.1° and 126.9°.
3How many solutions does sin x = 0.4 have from 0° to 720°? Explain using the graph.
In each 360° cycle, the line y = 0.4 crosses the curve twice. Two cycles give 4 solutions.
4A student graphed y = sin x on a calculator with x from 0 to 360 and saw a rapidly wiggling line. What is the likely problem?
The calculator is in radian mode, so it treats 360 as 360 radians (about 57 cycles). Switch to degree mode.
5Does any horizontal line meet the graph of y = sin x exactly once on 0° ≤ x < 360°? If so, which?
Yes: y = 1 (only at 90°) and y = −1 (only at 270°). Lines between −1 and 1 meet it twice; lines above 1 or below −1 never meet it.
Challenge
1Find all x from −360° to 360° with sin x = 1.
sin x = 1 at 90° and at 90° − 360° = −270°. So x = −270° and 90°.
2Explain why sin(180° − x) = sin x by using the unit circle.
The points for x and 180° − x are mirror images across the y-axis. A reflection in the y-axis changes the sign of the x-coordinate but keeps the y-coordinate (the height) the same, so the sines are equal.
3Without graphing, decide whether sin 1000° is positive or negative. Explain.
1000° − 2(360°) = 280°. So sin 1000° = sin 280°. 280° is in quadrant IV, so it is negative (sin 280° ≈ −0.985).
Quiz
1. Which point is on the graph of f(x) = sin x?
2. Why is f(x) = sin x a function?
3. sin 30° = 0.5. Which is also equal to 0.5?
4. How many values of x from 0° to 360° satisfy sin x = 0.2?
5. Using sin 40° ≈ 0.643, what is sin 220°?
6. A graphing calculator shows y = sin x as a fast-wiggling line on 0 to 360. What should you change?
Big idea: Building the sine function from the sine ratio (C2.3) and sketching its graph from 0° to 360° (start of C2.4).
How to help: Have your teen make the table of values with a calculator and plot it by hand on graph paper. Ask: "Where is the curve highest? Lowest? Where does it cross the axis?"
Common difficulties:
- Calculator in radian mode (sin 90 should give 1).
- Thinking a function cannot repeat outputs. It can; it just cannot give two outputs for one input.
- Drawing sharp points at the top and bottom. The sine curve is smooth and rounded at 90° and 270°.