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Question: If \( \sin 5\theta = \cos 4\theta \) , where \( 5\theta \) and \( 4\theta \) are acute angles , find...

If sin⁡5θ=cos⁡4θ\sin 5\theta = \cos 4\theta , where 5θ5\theta and 4θ4\theta are acute angles , find the values of θ\theta ?

Explanation

Solution

Start by writing the given equation and all the conditions given. Use the trigonometric identities used for the conversion of one trigonometric ratio to another one. After conversion substitute the value and compare the values and solve the relation formed then , in order to find out the value of θ\theta . Do a check on the value found by checking the range it can fall into.

Complete step-by-step answer:
Given,
sin⁡5θ=cos⁡4θ\sin 5\theta = \cos 4\theta
Also , 5θ5\theta and 4θ4\theta are acute angles.
Now, we know that sin⁡(90∘−θ)=cos⁡θ\sin ({90^ \circ } - \theta ) = \cos \theta
So , applying this identity for 4θ4\theta , we will get
sin⁡(90∘−5θ)=cos⁡4θ\sin ({90^ \circ } - 5\theta ) = \cos 4\theta
Substituting this values , we get
sin⁡5θ=sin⁡(90∘−4θ)\sin 5\theta = \sin ({90^ \circ } - 4\theta )
Now , we know that if sin⁡θ=sin⁡ϕ\sin \theta = \sin \phi then θ=ϕ\theta = \phi
∴5θ=90∘−4θ ⇒5θ+4θ=90∘ ⇒9θ=90∘ ⇒θ=90∘9 ∴θ=10∘  \therefore 5\theta = {90^ \circ } - 4\theta \\\ \Rightarrow 5\theta + 4\theta = {90^ \circ } \\\ \Rightarrow 9\theta = {90^ \circ } \\\ \Rightarrow \theta = \dfrac{{{{90}^ \circ }}}{9} \\\ \therefore \theta = {10^ \circ } \\\
So , The value of θ\theta is 10∘{10^ \circ } .
We can do a final check by using the data given in the question that is 5θ5\theta and 4θ4\theta are acute angles. So which means if we divide 5θ5\theta and 4θ4\theta by 5 and 4 respectively, we will get θ\theta also to acute , as
0⩽5θ⩽90∘ 0⩽5θ5⩽90∘5 0⩽θ⩽18∘  0 \leqslant 5\theta \leqslant {90^ \circ } \\\ 0 \leqslant \dfrac{{5\theta }}{5} \leqslant \dfrac{{{{90}^ \circ }}}{5} \\\ 0 \leqslant \theta \leqslant {18^ \circ } \\\
Which is also acute.

Note: Similar problems which involve different angles can be solved by following the same procedure. Attention must be given while converting one trigonometric ratio to another , by following the rules of conversion. A final check is must for such problems as sometimes we might get vague values, this can be done by either putting back the value in any given equation and look for expected value or by using the same procedure as in the solution above. In any case it must hold true.