Question

How many of the following trigonometric ratios are correct?

(1)Sin (${15}^{\circ }$) = $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(2)Cos (${105}^{\circ }$) = $\frac{1-\sqrt{3}}{2\sqrt{2}}$

(3)Sin ${75}^{\circ }$ = $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(4)Tan ${15}^{\circ }$ = $2-\sqrt{3}$

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Answer 1

The angles 15,105, and 75 are not standard angles. We know the trigonometric ratios of 30,45,60,90 etc. We also learned the formulae for finding

$sin(A\mp B),cos(A\mp B),tan(A\mp B)$ etc. we will solve this question using these two concepts / formulae.

For that, we should be able to express 15, 75 and 105 in terms of 0, 30, 45, 60 and 90, preferably as sum or difference of two angles. This part is easy to guess.

15 = 60 - 45 or 45 - 30

75 = 60 + 45 or 30 + 45

105 = 90 + 15 or 60 + 45

(1) Sin15 = sin(45 - 30)

= sin45 cos30 - sin30 cos 45

= $\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}-\frac{1}{2}\times \frac{1}{\sqrt{2}}$

= $\frac{\sqrt{3}-1}{2\sqrt{2}}$

(2) cos 105 = cos(90+15)

= - sin15

= - $\left(\frac{\sqrt{3}-1}{2\sqrt{2}}\right)$

= $\frac{1-\sqrt{3}}{2\sqrt{2}}$

(3) sin 75 = sin(45 + 30)

= sin45cos30 + sin30cos45

= $\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}+\frac{1}{2}\times \frac{1}{\sqrt{2}}$

= $\frac{\sqrt{3}+1}{2\sqrt{2}}$

(4) tan 15 = tan (45-30)

= $\frac{tan45-tan30}{1+tan45tan30}$

= $\frac{1-\frac{1}{\sqrt{3}}}{1+\frac{1}{\sqrt{3}}}$

= $\frac{\sqrt{3}-1}{\sqrt{3}+1}$

= $\frac{(\sqrt{3}-1{)}^2}{(\sqrt{3}+1)(\sqrt{3}-1)}$

= 2 - $\sqrt{3}$

Key steps/concepts: (1) Expressing the given angles in terms of angles whose trig ratios are known

(2) Formulae for compound angles.