How do you find exact value of \tan(\pi/4)?

Haven

Haven

Answered question

2021-10-25

How do you find exact value of tan(π4)?

Answer & Explanation

Macsen Nixon

Macsen Nixon

Skilled2021-10-26Added 117 answers

Given: tan(π4)
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.
sin(π4)=adjacent leghypotenuse
cos(π4)=opposite leghypotenuse
By the Pythagorean theorem a2+b2=c2 where a (adjacent) and b (opposite) are the legs and c is the hypotenuse c. This then implies that c=a2+b2 for the hypotenuse c.
sin(π4)=aa2+b2
cos(π4)=ba2+b2
However, if one of the angles of the triangle is 45 or π4, then the triangle is isosceles and thus the two legs have the same length a=b
cos(π4)=ba2+b2
=bb2+b2
=b2b2
=b2b2
=b2b
=12
=1222
=2(2)2
=22
sin(π4)=aa2+b2=ba2+b2=cos(π4)
The tangent is the sine divided by the cousine:
tan(π4)=sin(π4)cos(π4)=2222=1
Result: 1

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