TOPICS
Search

Trigonometry Angles--Pi/5


The trigonometric formulas for pi/5 can be derived using the multiple-angle formula

 sin(5theta)=5sintheta-20sin^3theta+16sin^5theta.
(1)

Letting theta=pi/5 and x=sintheta then gives

 sinpi=0=5x-20x^3+16x^5.
(2)

Factoring out one power of x gives

 16x^4-20x^2+5=0.
(3)

Solving the quadratic equation for x^2 gives

 x^2=1/8(5+/-sqrt(5)).
(4)

But x=sin(pi/5) must be less than

 sin(pi/4)=1/2sqrt(2),
(5)

so taking the minus sign and simplifying gives

 sin(pi/5)=sqrt((5-sqrt(5))/8)=1/4sqrt(10-2sqrt(5)).
(6)
TrigonometryAnglesPi5

Filling in the remainder of the trigonometry functions then gives

cos(pi/5)=1/4(1+sqrt(5))
(7)
cot(pi/5)=1/5sqrt(25+10sqrt(5))
(8)
csc(pi/5)=1/5sqrt(50+10sqrt(5))
(9)
sec(pi/5)=sqrt(5)-1
(10)
sin(pi/5)=1/4sqrt(10-2sqrt(5))
(11)
tan(pi/5)=sqrt(5-2sqrt(5))
(12)

and

cos((2pi)/5)=1/4(-1+sqrt(5))
(13)
cot((2pi)/5)=1/5sqrt(25-10sqrt(5))
(14)
csc((2pi)/5)=1/5sqrt(50-10sqrt(5))
(15)
sec((2pi)/5)=1+sqrt(5)
(16)
sin((2pi)/5)=1/4sqrt(10+2sqrt(5))
(17)
tan((2pi)/5)=sqrt(5+2sqrt(5)).
(18)

See also

Dodecahedron, Golden Ratio, Icosahedron, Pentagon, Pentagram, Trigonometry Angles, Trigonometry

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Trigonometry Angles--Pi/5." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/TrigonometryAnglesPi5.html

Subject classifications