 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-1/3/solution

Matching the two parametric equations at an intersection gives

cos ϕ = - t

and

sin ϕ = 1 + t

t^{2} + (1 + t)^{2} = 1, 2 t (t + 1) = 0.

For

t = 0

, the sine is one and the cosine is zero; for

t = - 1

, the cosine is one and the sine is zero. Keeping the stated half-open angular interval gives

(t, ϕ) = (0, π /2) or (- 1, 0) .

The corresponding points are

(0, 1)

and

(2, 0)

; both satisfy the original parametric equations.

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