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2021-10-25

How do you find exact value of $\mathrm{tan}\left(\frac{\pi}{4}\right)$ ?

Macsen Nixon

Skilled2021-10-26Added 117 answers

Given: $\mathrm{tan}\left(\frac{\pi}{4}\right)$

The cousine is defined as the opposite leg divided by the hypotenuse of a rectangular triangle, while the sine is defined as the adjacent leg divided by the hypotenuse.

$\mathrm{sin}\left(\frac{\pi}{4}\right)=\frac{\text{adjacent leg}}{\text{hypotenuse}}$

$\mathrm{cos}\left(\frac{\pi}{4}\right)=\frac{\text{opposite leg}}{\text{hypotenuse}}$

By the Pythagorean theorem$a}^{2}+{b}^{2}={c}^{2$ where a (adjacent) and b (opposite) are the legs and c is the hypotenuse c. This then implies that $c=\sqrt{{a}^{2}+{b}^{2}}$ for the hypotenuse c.

$\mathrm{sin}\left(\frac{\pi}{4}\right)=\frac{a}{\sqrt{{a}^{2}+{b}^{2}}}$

$\mathrm{cos}\left(\frac{\pi}{4}\right)=\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}$

However, if one of the angles of the triangle is$45}^{\circ$ or $\frac{\pi}{4}$ , then the triangle is isosceles and thus the two legs have the same length a=b

$\mathrm{cos}\left(\frac{\pi}{4}\right)=\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}$

$=\frac{b}{\sqrt{{b}^{2}+{b}^{2}}}$

$=\frac{b}{\sqrt{2{b}^{2}}}$

$=\frac{b}{\sqrt{2}\sqrt{{b}^{2}}}$

$=\frac{b}{\sqrt{2}b}$

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

$=\frac{1\cdot \sqrt{2}}{\sqrt{2}\cdot \sqrt{2}}$

$=\frac{\sqrt{2}}{{\left(\sqrt{2}\right)}^{2}}$

$=\frac{\sqrt{2}}{2}$

$\mathrm{sin}\left(\frac{\pi}{4}\right)=\frac{a}{\sqrt{{a}^{2}+{b}^{2}}}=\frac{b}{\sqrt{{a}^{2}+{b}^{2}}}=\mathrm{cos}\left(\frac{\pi}{4}\right)$

The tangent is the sine divided by the cousine:

$\mathrm{tan}\left(\frac{\pi}{4}\right)=\frac{\mathrm{sin}\left(\frac{\pi}{4}\right)}{\mathrm{cos}\left(\frac{\pi}{4}\right)}=\frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}}=1$

Result: 1

The cousine is defined as the opposite leg divided by the hypotenuse of a rectangular triangle, while the sine is defined as the adjacent leg divided by the hypotenuse.

By the Pythagorean theorem

However, if one of the angles of the triangle is

The tangent is the sine divided by the cousine:

Result: 1

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