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Question Number 143680 by mathlove last updated on 17/Jun/21

$$\tan 76=4$$$$\sin {}^{2}14=?$$

Commented by mr W last updated on 17/Jun/21

$$\tan 14=\frac{1}{\tan 76}=\frac{1}{a}$$$$\sin 14=\frac{1}{\sqrt{1^2+a^2}}=\frac{1}{\sqrt{1+a^2}}$$$${\sin }^{2}14=\frac{1}{1+a^2}$$$$a\approx 4$$$${\sin }^{2}14\approx \frac{1}{17}$$

Answered by TheHoneyCat last updated on 17/Jun/21

$$tan=\frac{sin}{cos}$$$$\iff {cos}^2{tan}^2={sin}^2$$$$\iff {cos}^2{tan}^2=1-{cos}^2$$$$\iff {cos}^2({tan}^2+1)=1$$$$\iff {cos}^2=\frac{1}{{tan}^2+1}$$$$sin\left(x\right)=cos(x-90)=cos(90-x)$$$$\mathrm{thus}:{sin}^2\left(x\right)={cos}^2(90-x)=\frac{1}{{tan}^2(90-x)+1}$$$$\mathrm{evaluating}\mathrm{for}x=14$$$${sin}^{2}14=\frac{1}{{tan}^2\left(76\right)+1}$$$${sin}^{2}14=\frac{1}{5}{}_{\mathrm{according}\mathrm{to}\mathrm{your}\mathrm{own}\mathrm{result}}$$$$\mathrm{Howerver}\mathrm{it}\mathrm{is}\mathrm{not}\mathrm{a}\mathrm{exact}\mathrm{value},$$$$tan76\approx 4,010780934...$$$$\mathrm{But}\mathrm{it}\mathrm{is}\mathrm{indeed}\mathrm{a}\mathrm{good}\mathrm{approximation}.$$$$\mathrm{and}\mathrm{make}\mathrm{a}\mathrm{fun}\mathrm{exercise};\bullet )$$

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