Goal: Understand that solving an equation is finding where the wave (blue) crosses a horizontal line (red).
Use the sliders to see how changing the wave or moving the target line affects the number and location of solutions (grey dots).
Formula: If $\sin(X) = c$, then $X = n\pi + (-1)^n \alpha$, where $\alpha = \sin^{-1}(c)$.
Example Steps: Solve $2\sin(x) = 1$
1. Isolate the trig function: $\sin(x) = 0.5$
2. Find Principal Value ($\alpha$): $\alpha = \sin^{-1}(0.5) = \frac{\pi}{6}$
3. Apply General Formula: $x = n\pi + (-1)^n(\frac{\pi}{6})$
4. Generate solutions for $n=0, 1, 2...$
Casio Graphing Calculator
1. Press SHIFT.
2. Press SIN to get $\sin^{-1}$.
Context: The depth of water $d$ (metres) at a port $t$ hours after midnight is modelled by: $$d(t) = 3\sin(0.5t) + 5$$
Your Task: Find the times when the water depth is exactly 6.5 metres.
Instructions: In the Desmos window below, type the two equations needed to solve this graphically:
y = 3sin(0.5x) + 5y = ? (You decide this value)Formula: If $\cos(X) = c$, then $X = 2n\pi \pm \alpha$, where $\alpha = \cos^{-1}(c)$.
Example Steps: Solve $2\cos(x) = \sqrt{3}$
1. Isolate the trig function: $\cos(x) = \frac{\sqrt{3}}{2}$
2. Find Principal Value ($\alpha$): $\alpha = \cos^{-1}(\frac{\sqrt{3}}{2}) = \frac{\pi}{6}$
3. Apply General Formula: $x = 2n\pi \pm \frac{\pi}{6}$
4. Generate solutions using both $+$ and $-$ for $n=0, 1, 2...$
Casio Graphing Calculator
1. Press SHIFT.
2. Press COS to get $\cos^{-1}$.
Context: The temperature $T$ (°C) in a greenhouse $h$ hours after 8:00 AM is modelled by: $$T(h) = 4\cos(\frac{\pi}{12}h) + 18$$
Your Task: Find the times when the temperature is exactly 20°C.
Instructions: In the Desmos window below, type the two equations needed to find the intersection:
y = 4cos(pi/12 * x) + 18 (Type 'pi' for $\pi$)y = ?