Trickonometry I

Geometry Level 3

$\sin \theta = \dfrac {\sqrt 6 + \sqrt 2}4$

Without using a calculator , which of the following values is a solution to $\theta$ for the equation above?

Note: This Trigonometric Ratios Table may help.

115^\circ

95^\circ

85^\circ

120^\circ

105^\circ

2 solutions

Ethan Mandelez
Apr 5, 2021

Recall the identity

$\sin (A+B) = \sin A \cos B + \cos A \sin B$

Now, let's re-write the fraction to see whether we can rearrange it into that form.

$\dfrac {\sqrt{6} + \sqrt{2}} {4} = \dfrac {\sqrt {6}} {4} + \dfrac {\sqrt {2}} {4}$

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

Since this is equivalent to:

$\sin 60 \cos 45 + \cos 60 \sin 45$

Therefore it is equal to $\sin (60+45)$ $= \sin 105$ .

If you haven't learnt Fundamental Trigonometric Identities and Specific Angles , I recommend to do so. They are very useful :D

Well I did it differently, the given value of sin = 0.96, so this immediately rules all options except sin(85) and sin(105). But since sin(85) = cos(5) and for 5 degree it said that cos will tend to 1 and 0.96 has a difference of 0.04 and 0.04 is huge in terms of approximation's. So the only possible answer is sin(105)

Omek K - 2 months, 1 week ago

you mean putting the options given in $\theta$ ? That's another good approach to this question 👌

Ethan Mandelez - 2 months, 1 week ago

Sorry I was editing. That's what I meant

Omek K - 2 months, 1 week ago

Yep, got it

Ethan Mandelez - 2 months, 1 week ago

Chew-Seong Cheong
Apr 5, 2021

$\begin{aligned} \sin \theta & = \frac {\sqrt 6 + \sqrt 2}4 \\ & = \frac {\sqrt 3}2 \cdot \frac {\sqrt 2}2 + \frac 12 \cdot \frac {\sqrt 2}2 \\ & = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \\ & = \sin (60^\circ + 45^\circ) \\ & = \sin (105^\circ) \\ \implies \theta & = \boxed{105^\circ} \end{aligned}$

Trickonometry I

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