07 Trigonometric functions · g10-07-01 · 5–10 min
What does one radian measure?
Does doubling a circle radius double its angle or its arc?
Predict: For a 90° turn, predict the arc lengths when r=1 and r=2.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Turn (degrees) = 90; Radius r = 1 | |
Case 2
Turn (degrees) = 90; Radius r = 2 | |
Case 3
Turn (degrees) = 0; Radius r = 2 | |
Revise: My claim needs the condition… because…
Exit: Find the arc of a 60° sector with radius 3.
07 Trigonometric functions · g10-07-02 · 5–10 min
One circle, two coordinates
Which coordinate is sine, and which signs depend on the quadrant?
Predict: Compare sine and cosine at 30°,150°,210° and330°.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Angle θ (degrees) = 30 | |
Case 2
Angle θ (degrees) = 150 | |
Case 3
Angle θ (degrees) = 270 | |
Revise: My claim needs the condition… because…
Exit: Give the coordinates at270°.
07 Trigonometric functions · g10-07-03 · 5–10 min
Reflect an angle without recalculating
Why do θ and180°−θ share a sine but not always a cosine?
Predict: Predict which coordinate changes under reflection in the y-axis.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Original angle θ (degrees) = 30 | |
Case 2
Original angle θ (degrees) = 120 | |
Case 3
Original angle θ (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: Express cos150° in terms of cos30°.
07 Trigonometric functions · g10-07-04 · 5–10 min
A square loses a sign
Does sin²θ+cos²θ=1 determine the sign of cosine?
Predict: Compare30° and150° before computing their squared coordinates.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Angle θ (degrees) = 30 | |
Case 2
Angle θ (degrees) = 150 | |
Case 3
Angle θ (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: If sinθ=1/2 and θ lies in quadrantII, find cosθ.
07 Trigonometric functions · g10-07-05 · 5–10 min
Unroll the circle into a wave
How does a moving circle point produce a sine graph?
Predict: Predict the graph height after a quarter, half and full turn.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Turn (degrees) = 90 | |
Case 2
Turn (degrees) = 180 | |
Case 3
Turn (degrees) = 360 | |
Revise: My claim needs the condition… because…
Exit: What graph point corresponds to one full turn?
07 Trigonometric functions · g10-07-06 · 5–10 min
Where does tangent break?
Can a tangent graph pass through90°?
Predict: Predict tanθ at85°,90°,95°.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Angle θ (degrees) = 85 | |
Case 2
Angle θ (degrees) = 90 | |
Case 3
Angle θ (degrees) = 95 | |
Revise: My claim needs the condition… because…
Exit: State whether tan270° exists.
07 Trigonometric functions · g10-07-07 · 5–10 min
Can sine distribute over addition?
Is sin(α+β)=sinα+sinβ?
Predict: Test α=β=30° mentally before revealing the result.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
α (degrees) = 30; β (degrees) = 30 | |
Case 2
α (degrees) = 30; β (degrees) = 60 | |
Case 3
α (degrees) = 30; β (degrees) = 0 | |
Revise: My claim needs the condition… because…
Exit: Give an exact counterexample to the additive claim.
07 Trigonometric functions · g10-07-08 · 5–10 min
Double an angle, not its sine
How does sin2θ relate to sinθ?
Predict: At θ=45°, compare sin2θ,2sinθ and2sinθcosθ.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Angle θ (degrees) = 45 | |
Case 2
Angle θ (degrees) = 30 | |
Case 3
Angle θ (degrees) = 0 | |
Revise: My claim needs the condition… because…
Exit: Find sin90° using θ=45° in the double-angle formula.
07 Trigonometric functions · g10-07-09 · 5–10 min
Amplitude and a reflected wave
Does a negative coefficient mean a negative amplitude?
Predict: Compare the ranges of2sinx,−2sinx and0sinx.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Coefficient A = 2 | |
Case 2
Coefficient A = -2 | |
Case 3
Coefficient A = 0 | |
Revise: My claim needs the condition… because…
Exit: State the amplitude and range of−3sinx.
07 Trigonometric functions · g10-07-10 · 5–10 min
Frequency squeezes the horizontal axis
Why is the period2π/|ω| rather than2πω?
Predict: Compare sinx,sin2x andsin(−2x) on the same interval.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Frequency ω = 1 | |
Case 2
Frequency ω = 2 | |
Case 3
Frequency ω = 0 | |
Revise: My claim needs the condition… because…
Exit: Find the basic period ofsin(3x).
07 Trigonometric functions · g10-07-11 · 5–10 min
Phase is divided by frequency
Does adding φ shift every sine graph by−φ?
Predict: Compare sin(x+90°) andsin(2x+90°), using x in degrees for this comparison.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Frequency ω = 1; Phase φ (degrees) = 90 | |
Case 2
Frequency ω = 2; Phase φ (degrees) = 90 | |
Case 3
Frequency ω = 0; Phase φ (degrees) = 90 | |
Revise: My claim needs the condition… because…
Exit: Find the shift ofsin(2x+π/2).
07 Trigonometric functions · g10-07-12 · 5–10 min
A wheel needs a starting position
Can amplitude and period alone determine a height model?
Predict: A wheel has radius5, centre height6 and period40s. Predict heights at0,10 and30s when starting on the right and moving upwards.
Case numbers match the projection buttons. Record the displayed conditions and your observation.
| Compare | What changed? What stayed? |
|---|
Case 1
Radius R (m) = 5; Centre height H (m) = 6; Period T (s) = 40; Time t (s) = 10; Starting phase (degrees) = 0 | |
Case 2
Radius R (m) = 5; Centre height H (m) = 6; Period T (s) = 40; Time t (s) = 30; Starting phase (degrees) = 0 | |
Case 3
Radius R (m) = 5; Centre height H (m) = 6; Period T (s) = 0; Time t (s) = 10; Starting phase (degrees) = 0 | |
Revise: My claim needs the condition… because…
Exit: For the stated wheel, find h(10) andh(30).
