1.9: Exact trigonometric values and an addition formula

4PM1/1/June/2021 — Question 2 · 6 marks

Angle α\displaystyle \alpha is acute such that cos α=35\displaystyle \mathrm{cos}\,\alpha=\frac35.
Angle β\displaystyle \beta is obtuse such that sin β=12\displaystyle \mathrm{sin}\,\beta=\frac12.
(a) Find the exact value of
(i) tan α\displaystyle \mathrm{tan}\,\alpha,
(ii) tan β\displaystyle \mathrm{tan}\,\beta.
(3)
(b) Hence show that
tan(α+β)=m3−nn3+m,\mathrm{tan}(\alpha+\beta)=\frac{m\sqrt3-n}{n\sqrt3+m},
where m\displaystyle m and n\displaystyle n are positive integers whose values are to be found.
(3)