At a glance. One lesson of about 60 minutes. It assumes sine, cosine and tangent in right-angled triangles and the exact values for 30, 45 and 60 degrees from Unit 1. No calculator is needed; the second figure sets up the symmetry that lesson 4 uses to solve equations.
Calculator: no calculator. All the values here are exact or given, so no calculator is needed. Students who reach for one should be asked which quadrant the angle is in first.
What students will learn
Know
- The point on the circle of radius 1 at angle θ is (cos θ, sin θ).
- Which of sin, cos and tan is positive in each quadrant.
- tan θ = sin θ ÷ cos θ.
Understand
- The right-angled triangle definition is the case of an acute angle on the circle.
- The reference angle gives the size of the value and the quadrant gives the sign.
- Angles with a common reference angle are mirror images in the axes.
Do
- State the sign of sin, cos and tan for any angle.
- Find exact values for angles in all four quadrants using reference angles.
- Find the other angles between 0 and 360 degrees with the same sine, cosine or tangent.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: the definition, the signs and reference angles (check and questions 1 to 8).
- Criterion B. Investigating patterns: the figures. Ask for a prediction of the sign pattern before the slider moves, and the pattern in the four angles of the second figure.
- Criterion C. Communicating: giving the quadrant, the reference angle and the sign as three separate written steps.
- Criterion D. Applying mathematics in real-life contexts: the turning pedal and crank, where the height changes sign as it passes the axle.
The one thing to do with the figure
Make students commit before they slide. In the first figure, ask “Without moving it, is the cosine positive or negative at 100°? Is the tangent?” and take votes before the slider moves.
In the second figure, ask for the other three angles before they appear. Many give only 180° − r and miss the two below the axis.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 7 min | Hook: the turning pedal. Collect predictions of above or below, ahead or behind. |
| 10 min | First figure and key idea: signs by quadrant. |
| 10 min | Second figure, the warning card and the worked examples; pairs finish the faded example. |
| 5 min | The check. If many choose A, ask which quadrant 248° is in. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | √2 ÷ 2; second quadrant; 0.5. |
|---|---|
| Hook | Above the axle and behind it: sin positive, cos negative. |
| Worked examples | sin 135° = √2 ÷ 2; cos 210° = −√3 ÷ 2. |
| Faded example | tan 300° = −√3. |
| 1. Signs in a quadrant | D. A is the second quadrant; B is the fourth; C only holds up to 90°. |
| 2. Cosine in the second quadrant | −0.37. −cos 68°. |
| 3. A second angle with the same sine | 157. 180 − 23. |
| 4. A second angle with the same tangent | 208. 180 + 28. |
| 5. Tangent from a point on the circle | −1.33. 0.8 ÷ (−0.6). |
| 6. Height of a rotating tip | −15.6. 18 × (−0.866). |
| 7. How many angles share a sine | 1. Only 270°. |
| 8. Which angle has these signs | B. A is first quadrant; C has positive sin; D has positive tan. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or C in question 1 (the check) | Remembers one quadrant's pattern, or thinks all angles under 360 give positive values. | Ask: is the tip above or below the axis? Left or right of the vertical axis? |
| 0.37, −0.93 or 0.93 in question 2 | Keeps the acute angle's sign, uses sine for cosine, or reads the height. | Say aloud: cosine is across, sine is up. |
| 337, 203 or 67 in question 3 | Reflects in the wrong axis, or uses the complement. | Sketch the circle and the horizontal line at height 0.39. |
| 152, 332 or 62 in question 4 | Uses the sine pattern for the tangent, or the complement. | Tan is positive in the first and third quadrants, so add 180. |
| −0.75, 1.33 or 0.75 in question 5 | Divides cos by sin, or drops the sign of the cosine. | Write sin ÷ cos with the signed values in brackets. |
| 15.6, −9 or 9 in question 6 | Drops the sign, or uses the cosine. | Ask: is the height the sine or the cosine? |
| 2, 0 or 4 in question 7 | Applies the two-angle pattern where the line only touches the circle, misses the lowest point, or counts one per quadrant. | Draw the horizontal line at height −1 and ask whether it crosses or touches. |
| A, C or D in question 8 | Takes the first quadrant, or mixes the sign of sin with that of tan. | Fill a table of the signs of sin, cos, tan for each quadrant. |
Questions worth asking
- Why does the radius of length 1 make the coordinates equal the sine and cosine? In a right-angled triangle with hypotenuse 1, the opposite side is sin and the adjacent side is cos, with nothing to divide by.
- Where is tan not defined, and why? At 90 and 270 degrees, where cos is 0 and dividing by it is not possible.
- How are 40, 140, 220 and 320 degrees related? They share the reference angle 40 degrees and are mirror images in the axes.
- Does a bigger angle mean a bigger sine? No: the sine rises to 1 at 90 degrees, then falls again.
Differentiation
A way in for anyone stuck
- Give a printed circle with the four quadrants labelled and ask students to write the signs of sin, cos and tan in each.
- Use a three-step frame on the board: quadrant, reference angle, sign.
- Offer the exact values of 30, 45 and 60 degrees as a small card.
Stretch
- Show that sin² θ + cos² θ = 1 for every point on the circle, using Pythagoras.
- What are sin, cos and tan of 450°? Say what angle between 0° and 360° it matches and why.
Students learning in English as an additional language
“Anticlockwise” means the direction opposite to the hands of a clock. “Quadrant” is one quarter of the plane. “Reference angle” is the acute angle between the radius and the horizontal axis. “Undefined” means there is no value.
| Word | Plain meaning | Example |
|---|---|---|
| unit circle | a circle of radius 1 with its centre at the origin | the circle the crank traces |
| quadrant | one of the four quarters of the coordinate plane | the second quadrant is top left |
| reference angle | the acute angle between the radius and the horizontal axis | the reference angle of 100° is 80° |
| anticlockwise | the direction opposite to a clock's hands, in which angles are measured | from the positive x-axis upward |
| undefined | has no value because it needs a division by zero | tan at 270° |
Sentence starters for explaining aloud or in writing:
- The angle is in the … quadrant, so sin is ….
- The reference angle is …, so the size is ….
- The other angle with the same sine is …, because ….
Two things not to say
“Just use the calculator.” The quadrant and the reference angle are the idea.
“Sine is always positive.” That is true only above the horizontal axis.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- SignsIs cos 100° positive or negative?
Negative: the tip is left of the vertical axis. - ReferenceState the reference angle of 330°.
30°. 360 − 330.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- Exact valueFind the exact value of cos 150°.
−√3 ÷ 2. Second quadrant, reference angle 30°. - Earlier work (Unit 1)A right-angled triangle has legs 5 and 12. Find the hypotenuse.
13.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningCan cos θ = 0.5 and sin θ = 0.5 for the same angle? Use the circle.
No: 0.25 + 0.25 = 0.5, not 1, so the point is not on the circle. - Working backwardsGive two angles between 0° and 360° whose cosine is −0.7071.
135° and 225°.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both sliders work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each figure, “Show me the steps” gives the same states in words.
No device? Draw a circle of radius 10 cm on squared paper and ask pairs to measure the height and sideways distance at several angles.