"Find the value of cos18∘" was answered by our community

Rory Moran

Answered question

2023-02-22

Find the value of

\cos 18^{\circ}

Samantha Summers

Beginner2023-02-23Added 8 answers

Find the value of

\cos 18 °

using multiple angle formulae of trigonometric ratios
Let angle

A = 18 °

Thus,

5 A = 90 °

\Rightarrow 2 A + 3 A = 90 ˚

\Rightarrow

2 A = 90 ˚ - 3 A

Taking sin e on both sides, we get

\sin 2 A = \sin (90 ˚ - 3 A) = \cos 3 A

\Rightarrow

2 \sin A \times \cos A = 4 \cos^{3} A - 3 \cos A

\Rightarrow 2 \sin A \times \cos A - 4 \cos^{3} A + 3 \cos A = 0

\Rightarrow

\cos A (2 \sin A - 4 \cos^{2} A + 3) = 0

Dividing both sides by

\cos A = \cos 18 ˚ \neq 0

, we get

2 \sin A - 4 (1 - \sin^{2} A) + 3 = 0

\Rightarrow 4 \sin^{2} A + 2 \sin A - 1 = 0

, which is a quadratic equation in

\sin A

\Rightarrow \sin A = \frac{- 2 \pm {(2)}^{2} - 4 (4) (- 1)}{2 (4)}

= \frac{- 2 \pm 16 + 4}{8} = \frac{- 2 \pm 25}{8}

\Rightarrow

\sin A = \frac{- 1 \pm 5}{4}

Now the value of

\cos 18^{\circ}

\Rightarrow

\cos 18^{\circ} = 1 - \sin^{2} 18^{\circ}

= 1 - {(\frac{5 - 1}{4})}^{2} = 1 - (\frac{5 + 1 - 25}{16}) = \frac{10 + 25}{16}

\Rightarrow

\cos 18^{\circ} = \frac{10}{+}

Thus, the value of

\cos 18^{\circ} = \frac{10}{+}

Do you have a similar question?

Recalculate according to your conditions!

Didn't find what you were looking for?