D. Trigonometric Functions · Sinusoidal FunctionsFrom the Sine Ratio to the Sine Function
D2.3About 70 minutes
Learning goals- I can use the unit circle to find sin θ and cos θ for angles from 0° to 360°.
- I can make a table of values and graph y = sin θ and y = cos θ for 0° ≤ θ ≤ 360°.
- I can explain why y = sin θ and y = cos θ are functions.
- I can find sine and cosine of angles greater than 360° and of negative angles.
Words to know
unit circle
The circle with centre (0, 0) and radius 1. The point at angle θ on it is (cos θ, sin θ).
standard position
An angle drawn with its vertex at the origin and its initial arm on the positive x-axis, turning counterclockwise for positive angles.
terminal arm
The arm of an angle in standard position where the rotation ends.
coterminal angles
Angles that share the same terminal arm, such as 30°, 390° and −330°. They differ by a multiple of 360°.
related acute angle
The acute angle between the terminal arm and the x-axis.
sine function
The function y = sin θ that pairs each angle θ with its sine.
A ratio becomes a function
In Trigonometric Ratios and Laws you found sin 150° and cos 240° using a point on the terminal arm and the related acute angle. Each answer was a single number. Now we ask a bigger question: what happens to sin θ as θ keeps turning? If we list sin θ for many angles and plot the results, a beautiful wave appears. That wave is the graph of the sine function.
Remember from Trigonometric Ratios and Laws
For an angle θ in standard position with a point P(x, y) on its terminal arm, r = √(x² + y²) and
- sin θ = y/r, cos θ = x/r, tan θ = y/x
- Signs by quadrant (CAST): all ratios positive in quadrant 1; only sine in quadrant 2; only tangent in quadrant 3; only cosine in quadrant 4.
- Exact values: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2 (and cos 60° = 1/2, cos 45° = √2/2, cos 30° = √3/2).
The unit circle
Choose the point P where the terminal arm meets the circle of radius 1, the unit circle. Then r = 1, so
sin θ = y and cos θ = x.
In other words, the point at angle θ on the unit circle is P(cos θ, sin θ). The sine of an angle is simply the height of that point above the x-axis (negative if it is below), and the cosine is how far it is to the right (negative if it is to the left).
Making a table of values
As P travels counterclockwise once around the circle, its height starts at 0, rises to 1 at 90°, falls back to 0 at 180°, sinks to −1 at 270° and returns to 0 at 360°. Using exact values and related acute angles, we can fill in a table (decimals rounded to two places).
| θ | 0° | 30° | 60° | 90° | 120° | 150° | 180° | 210° | 240° | 270° | 300° | 330° | 360° |
|---|
| sin θ | 0 | 0.5 | 0.87 | 1 | 0.87 | 0.5 | 0 | −0.5 | −0.87 | −1 | −0.87 | −0.5 | 0 |
| cos θ | 1 | 0.87 | 0.5 | 0 | −0.5 | −0.87 | −1 | −0.87 | −0.5 | 0 | 0.5 | 0.87 | 1 |
Worked exampleExample 1: Exact values for the table
Find the exact values of sin 210° and cos 210°, and explain where they appear on the graph.
- 210° is in quadrant 3. The related acute angle is 210° − 180° = 30°.
- In quadrant 3 both x and y are negative, so sine and cosine are both negative.
- sin 210° = −sin 30° = −1/2. cos 210° = −cos 30° = −√3/2 (about −0.87).
- On the graph of y = sin θ, the point (210°, −0.5) is just after the graph crosses the axis at 180° and heads down toward its minimum at 270°.
sin 210° = −1/2 and cos 210° = −√3/2.
Why is y = sin θ a function?
A function gives exactly one output for each input. For each angle θ there is exactly one terminal arm, so exactly one point P on the unit circle, so exactly one height. Therefore each θ gives exactly one value of sin θ: y = sin θ is a function. Its graph passes the vertical line test. The same argument (using the x-coordinate) shows that y = cos θ is a function.
Compare this with the unit circle itself, x² + y² = 1. The vertical line x = 0.5 meets the circle twice, so the circle is not a function.
Notice that the sine function is not one-to-one: different angles can share a sine (sin 30° = sin 150°). That is why, in Trigonometric Ratios and Laws, you usually found two angles between 0° and 360° with the same ratio.
Going around again, and going backwards
Angles do not have to stop at 360°. An angle of 390° is one full turn plus 30°. It ends on the same terminal arm as 30°, so sin 390° = sin 30° = 0.5. Angles that share a terminal arm are coterminal; they differ by multiples of 360°.
A negative angle turns clockwise. The angle −90° ends on the negative y-axis, so sin(−90°) = −1.
Because any angle at all has a terminal arm, the domain of y = sin θ is all real numbers, and because θ and θ + 360° are coterminal, the pattern repeats: sin(θ + 360°) = sin θ. The sine function is periodic with period 360°.
Worked exampleExample 2: Big and negative angles
Find the exact value: (a) sin 405° (b) sin(−90°) (c) cos 720° (d) sin(−210°)
- (a) 405° − 360° = 45°, so sin 405° = sin 45° = √2/2.
- (b) −90° is a quarter turn clockwise, ending on the negative y-axis at (0, −1). So sin(−90°) = −1.
- (c) 720° is two full turns, coterminal with 0°. cos 720° = cos 0° = 1.
- (d) Add 360°: −210° + 360° = 150°, so sin(−210°) = sin 150° = 1/2.
Just for funThe word sine comes from the Latin sinus, meaning a fold or a bay. Historians trace it back through Arabic to a Sanskrit word for a bowstring. According to the usual story, a medieval translator read the Arabic word as a similar-looking word meaning a fold or pocket, and translated it as sinus. If so, one of the most important words in math is partly a translation mix-up!
Hands-on activityUnwrap a circle by hand
You need: a ruler, a protractor, a compass (or a round lid with a 10 cm radius traced carefully), a pencil, graph paper and a calculator.
- Draw a circle with radius 10 cm and draw x- and y-axes through its centre.
- With the protractor, mark points on the circle every 30°, from 0° to 330°, measured counterclockwise from the positive x-axis.
- For each point, measure its height above the x-axis in centimetres (write a negative number if it is below). Divide by 10. This is sin θ, because the radius is 10.
- Record your values in a table and compare them with your calculator (degree mode).
- On graph paper, plot (θ, your value) for θ = 0°, 30°, …, 360° and join the points with a smooth curve.
What to look for: your values should be close to the table in this lesson (for example, about 0.5 at 30° and about 0.87 at 60°). The graph is a smooth wave with a high point of 1 at 90° and a low point of −1 at 270°.
Your turn
1Write the coordinates of the points on the unit circle at 90°, 180° and 270°. Use them to state sin 90°, sin 180° and sin 270°.
(0, 1), (−1, 0), (0, −1). So sin 90° = 1, sin 180° = 0, sin 270° = −1.
2Find the exact values of sin 120°, sin 240° and sin 300°.
Each has related acute angle 60°. Sine is positive in quadrant 2 and negative in quadrants 3 and 4: √3/2, −√3/2, −√3/2.
3Find the exact values of sin 390° and sin 495°.
390° − 360° = 30°: 1/2. 495° − 360° = 135° (quadrant 2, related angle 45°): √2/2.
4Find the exact value of sin(−30°).
−30° is coterminal with 330° (quadrant 4), so sin(−30°) = −sin 30° = −1/2.
5Explain why the sine function has a period of 360°.
Adding 360° makes one full turn, which brings the terminal arm back to the same place. The point on the unit circle is the same, so the sine is the same: sin(θ + 360°) = sin θ.
6Is y = sin θ a one-to-one function? Find two angles between 0° and 360° with sin θ = −0.5.
No, it is not one-to-one. sin θ = −0.5 at θ = 210° and θ = 330° (related angle 30°, in quadrants 3 and 4).
7Complete: cos 0°, cos 90°, cos 180°, cos 270°, cos 360°.
1, 0, −1, 0, 1 (the x-coordinates of (1, 0), (0, 1), (−1, 0), (0, −1), (1, 0)).
More practice: warm-up
1On the unit circle, sin θ equals which coordinate of the point at angle θ?
2State sin 0° and sin 90°.
4What is the greatest value sin θ can have, and at which angle between 0° and 360°?
5Name a positive angle coterminal with 30°.
More practice: on your own
1Use a calculator to find sin 20°, sin 160°, sin 200° and sin 340° to two decimal places. Explain the pattern.
0.34, 0.34, −0.34, −0.34. All four have related acute angle 20°. Sine is positive in quadrants 1 and 2 and negative in quadrants 3 and 4, so only the signs change.
2Find the exact values of sin 225° and cos 225°.
Quadrant 3, related angle 45°, both negative: sin 225° = cos 225° = −√2/2.
3Find all angles between 0° and 360° with sin θ = √3/2.
Related angle 60°; sine positive in quadrants 1 and 2: θ = 60° and θ = 120°.
4Find the exact values of sin 750° and cos(−120°).
750° − 2(360°) = 30°, so sin 750° = 1/2. −120° + 360° = 240° (quadrant 3, related angle 60°), so cos(−120°) = −1/2.
5The point P(−0.6, 0.8) is on the unit circle at angle θ, where 0° ≤ θ ≤ 360°. State sin θ and cos θ, and find θ to the nearest tenth of a degree.
sin θ = 0.8, cos θ = −0.6. P is in quadrant 2. Related angle: sin⁻¹(0.8) ≈ 53.1°, so θ ≈ 180° − 53.1° = 126.9°.
Challenge
1Use the unit circle to explain why sin(θ + 180°) = −sin θ for every angle θ.
Adding 180° turns the terminal arm by half a turn, so it points in the exact opposite direction. The point (x, y) moves to (−x, −y). Its height changes from y to −y, so sin(θ + 180°) = −sin θ.
2For 0° ≤ θ ≤ 360°, how many solutions does sin θ = k have when (a) k = 1, (b) k = 0, (c) k = 0.3, (d) k = 1.2? Explain using the graph.
(a) One (θ = 90°): the line y = 1 touches the graph only at the maximum. (b) Three (0°, 180°, 360°). (c) Two: the line y = 0.3 cuts the hump between 0° and 180° twice. (d) None: the graph never goes above 1.
3A point on a circle of radius 5 centred at the origin has y-coordinate −3 and lies in quadrant 3. Find sin θ, cos θ and θ (to the nearest tenth of a degree).
sin θ = y/r = −3/5. x² + 9 = 25 gives x = −4 (quadrant 3), so cos θ = −4/5. Related angle sin⁻¹(3/5) ≈ 36.9°, so θ ≈ 180° + 36.9° = 216.9°.
Quiz
1. On the unit circle, the point on the terminal arm of angle θ is…
3. Why is y = sin θ a function?
5. What is the period of y = sin θ?
Big idea: D2.3 asks students to connect the sine and cosine ratios they already know to the sine and cosine functions, by graphing the ratios for 0° to 360° and explaining why the relationships are functions.
How to help: Ask your teen to show you the unit circle and point to where sin θ is (the height of the point). Turn a clock face or a bicycle wheel slowly and ask, "Is the height going up or down right now? Is it positive or negative?"
Common difficulties:
- Mixing up which coordinate is sine. Sine is the y (height); cosine is the x.
- Forgetting signs in quadrants 2, 3 and 4. A quick sketch of the terminal arm settles it.
- Calculator in radian mode. sin 30 should give 0.5; if it gives −0.988…, switch to degrees.
- Thinking "function" means "one-to-one". A function needs one output per input; different inputs may share an output.