Find Two Times at a Given Level

Supplied materials

• Paper or a note app.
• Fictional rule h(t) = 5 + 3cos(πt/6) for 0 ≤ t < 12. Supplied fact: in 0 ≤ θ < 2π, cos(θ)=1/2 only at θ=π/3 and θ=5π/3. Also cos(θ)=0 at π/2 and 3π/2 in that cycle.

Learn the idea

A level can occur twice in a cosine cycle, once on the way down and once on the way up. To find when h(t)=6.5, subtract the midline five and divide by amplitude three. The resulting equation is cos(πt/6)=1/2.

Use every supplied angle solution in the stated cycle, then solve the linear angle-to-time equation. The interval 0 ≤ t < 12 prevents adding solutions from later cycles. A single successful substitution is not enough to show that all times were found.

1. Rewrite 5 + 3cos(πt/6) = 6.5 as cosine = 1/2.
2. Set πt/6 equal to π/3 and then to 5π/3.
3. Solve for t, check both substitutions, and retain times in 0 ≤ t < 12.

Check your reasoning

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Take it further

Change the period to 6 by using r(t) = 5 + 3cos(πt/3), with 0 ≤ t < 6. The same level 6.5 now occurs at t = 1 and t = 5, because those times give the two supplied half-value angles.

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