syms f x y;
f=sin(x);
x0=0:2*pi/100:2*pi;
y=subs(diff(f,x),x0).*(x-x0) + subs(f,x0);
a=subs(y,x,1)-subs(y,x,0);
p=1./sqrt(1+a.^2);
q=acos(p);
pretty(q)
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Define equation of tangent line f=sin(x) at multiple point inputs
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Hi Matlab experts,
I would like to build a script to define the equation of tangent line of function f(x)=sin(x) at x0 (x0 = 0 : 2*pi/100 : 2*pi).
Then calculate its value at x=1 & x =0 ---> calculate p & q as below. But I failed to run it. Please help me complete the script. Thanks a lot!
syms f x y;
f=sin(x);
x0=0:2*pi/100:2*pi;
y=subs(diff(f,x),x0)*(x-x0) + subs(f,x0);
a=subs(y,x,1)-subs(y,x,0);
p=1/sqrt(1+a^2);
q=acos(p);
Answers (1)
VBBV
on 18 Apr 2021
Edited: VBBV
on 18 Apr 2021
4 Comments
Jack Dong
on 18 Apr 2021
Thanks VBBV but I would like to get the numberic result of q. Since I will use them for following calculation. Here I have 101 input value of x0 -> Hopefully I have 101 numberic results of q. Could you help me with those info.
Walter Roberson
on 18 Apr 2021
syms f x y;
f=sin(x);
x0=0:2*pi/100:2*pi;
y=subs(diff(f,x),x0).*(x-x0) + subs(f,x0);
a=subs(y,x,1)-subs(y,x,0);
p=1./sqrt(1+a.^2);
q=acos(p);
pretty(q)
-- -- pi / 1 \ / 1 \ / 1 \
| | --, acos| ---------------------------------------------------- |, acos| ---------------------------------------------------- |, acos| ------------------------------------------ |,
| | 4 | / / / pi \ \2 \ | | / / / pi \ \2 \ | | / / / 3 pi \ pi #9 3 \2 \ |
| | | | | pi cos| -- | | | | | | | pi cos| -- | | | | | sqrt| | #9 | ---- - 1 | - ------- | + 1 | |
| | | | | / pi \ / pi \ \ 50 / | | | | | | / pi \ / pi \ \ 25 / | | | \ \ \ \ 50 / 50 / / /
| | | sqrt| | cos| -- | | -- - 1 | - ------------ | + 1 | | | sqrt| | cos| -- | | -- - 1 | - ------------ | + 1 | |
-- -- \ \ \ \ 50 / \ 50 / 50 / / / \ \ \ \ 25 / \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ------------------------------------------------------------------------------------ |,
| / / / 2 pi \ pi #18 2 \2 \ | | / / / pi \ \2 \ |
| sqrt| | #18 | ---- - 1 | - -------- | + 1 | | | | | sqrt(2) | -- - 1 | sqrt(sqrt(5) + 5) | | |
\ \ \ \ 25 / 25 / / / | | | pi sqrt(2) sqrt(sqrt(5) + 5) \ 10 / | | |
| sqrt| | ---------------------------- - ------------------------------------ | + 1 | |
\ \ \ 40 4 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ------------------------------------------ |, acos| -------------------------------------------- |,
| / / / 3 pi \ pi #17 3 \2 \ | | / / / 7 pi \ pi #8 7 \2 \ | | / / / 4 pi \ pi #16 4 \2 \ |
| sqrt| | #17 | ---- - 1 | - -------- | + 1 | | | sqrt| | #8 | ---- - 1 | - ------- | + 1 | | | sqrt| | #16 | ---- - 1 | - -------- | + 1 | |
\ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \ / 1 \
acos| ------------------------------------------ |, acos| ---------------------------------------- |, acos| -------------------------------------------- |,
| / / / 9 pi \ pi #7 9 \2 \ | | / / / pi \ pi #20 \2 \ | | / / / 11 pi \ pi #6 11 \2 \ |
| sqrt| | #7 | ---- - 1 | - ------- | + 1 | | | sqrt| | | -- - 1 | #20 - ------ | + 1 | | | sqrt| | #6 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 6 pi \ pi #15 6 \2 \ | | / / / 13 pi \ pi #5 13 \2 \ | | / / / 7 pi \ pi #14 7 \2 \ |
| sqrt| | #15 | ---- - 1 | - -------- | + 1 | | | sqrt| | #5 | ----- - 1 | - -------- | + 1 | | | sqrt| | #14 | ---- - 1 | - -------- | + 1 | |
\ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ---------------------------------------------------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 3 pi \ \2 \ | | / / / 8 pi \ pi #13 8 \2 \ |
| | | sqrt(2) | ---- - 1 | sqrt(5 - sqrt(5)) | | | | sqrt| | #13 | ---- - 1 | - -------- | + 1 | |
| | | \ 10 / sqrt(2) pi sqrt(5 - sqrt(5)) 3 | | | \ \ \ \ 25 / 25 / / /
| sqrt| | -------------------------------------- - ------------------------------ | + 1 | |
\ \ \ 4 40 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 17 pi \ pi #4 17 \2 \ | | / / / 9 pi \ pi #12 9 \2 \ | | / / / 19 pi \ pi #3 19 \2 \ |
| sqrt| | #4 | ----- - 1 | - -------- | + 1 | | | sqrt| | #12 | ---- - 1 | - -------- | + 1 | | | sqrt| | #3 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 2 pi \ pi #19 2 \2 \ | | / / / 21 pi \ pi #2 21 \2 \ | | / / / 11 pi \ pi #11 11 \2 \ |
| sqrt| | | ---- - 1 | #19 - -------- | + 1 | | | sqrt| | #2 | ----- - 1 | - -------- | + 1 | | | sqrt| | #11 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, 0, acos| ---------------------------------------------- |,
| / / / 23 pi \ pi #1 23 \2 \ | | / / / 12 pi \ pi #10 12 \2 \ | | / / / 13 pi \ pi #10 13 \2 \ |
| sqrt| | #1 | ----- - 1 | - -------- | + 1 | | | sqrt| | #10 | ----- - 1 | - --------- | + 1 | | | sqrt| | #10 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 27 pi \ pi #1 27 \2 \ | | / / / 14 pi \ pi #11 14 \2 \ | | / / / 29 pi \ pi #2 29 \2 \ |
| sqrt| | #1 | ----- - 1 | - -------- | + 1 | | | sqrt| | #11 | ----- - 1 | - --------- | + 1 | | | sqrt| | #2 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 3 pi \ pi #19 3 \2 \ | | / / / 31 pi \ pi #3 31 \2 \ | | / / / 16 pi \ pi #12 16 \2 \ |
| sqrt| | | ---- - 1 | #19 - -------- | + 1 | | | sqrt| | #3 | ----- - 1 | - -------- | + 1 | | | sqrt| | #12 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 33 pi \ pi #4 33 \2 \ | | / / / 17 pi \ pi #13 17 \2 \ |
| sqrt| | #4 | ----- - 1 | - -------- | + 1 | | | sqrt| | #13 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ---------------------------------------------------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 7 pi \ \2 \ | | / / / 18 pi \ pi #14 18 \2 \ |
| | | sqrt(2) | ---- - 1 | sqrt(5 - sqrt(5)) | | | | sqrt| | #14 | ----- - 1 | - --------- | + 1 | |
| | | \ 10 / sqrt(2) pi sqrt(5 - sqrt(5)) 7 | | | \ \ \ \ 25 / 25 / / /
| sqrt| | -------------------------------------- - ------------------------------ | + 1 | |
\ \ \ 4 40 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 37 pi \ pi #5 37 \2 \ | | / / / 19 pi \ pi #15 19 \2 \ | | / / / 39 pi \ pi #6 39 \2 \ |
| sqrt| | #5 | ----- - 1 | - -------- | + 1 | | | sqrt| | #15 | ----- - 1 | - --------- | + 1 | | | sqrt| | #6 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 4 pi \ pi #20 4 \2 \ | | / / / 41 pi \ pi #7 41 \2 \ | | / / / 21 pi \ pi #16 21 \2 \ |
| sqrt| | | ---- - 1 | #20 - -------- | + 1 | | | sqrt| | #7 | ----- - 1 | - -------- | + 1 | | | sqrt| | #16 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 43 pi \ pi #8 43 \2 \ | | / / / 22 pi \ pi #17 22 \2 \ |
| sqrt| | #8 | ----- - 1 | - -------- | + 1 | | | sqrt| | #17 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ---------------------------------------------------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 9 pi \ \2 \ | | / / / 23 pi \ pi #18 23 \2 \ |
| | | sqrt(2) | ---- - 1 | sqrt(sqrt(5) + 5) | | | | sqrt| | #18 | ----- - 1 | - --------- | + 1 | |
| | | 9 pi sqrt(2) sqrt(sqrt(5) + 5) \ 10 / | | | \ \ \ \ 25 / 25 / / /
| sqrt| | ------------------------------ - -------------------------------------- | + 1 | |
\ \ \ 40 4 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------------------- |, acos| ---------------------------------------------------------- |,
| / / / 47 pi \ pi #9 47 \2 \ | | / / / pi \ \2 \ | | / / / pi \ \2 \ |
| sqrt| | #9 | ----- - 1 | - -------- | + 1 | | | | | pi cos| -- | 24 | | | | | | pi cos| -- | 49 | | |
\ \ \ \ 50 / 50 / / / | | | / pi \ / 24 pi \ \ 25 / | | | | | | / pi \ / 49 pi \ \ 50 / | | |
| sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | | | sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | |
\ \ \ \ 25 / \ 25 / 25 / / / \ \ \ \ 50 / \ 50 / 50 / / /
pi / 1 \ / 1 \
--, acos| ---------------------------------------------------------- |, acos| ---------------------------------------------------------- |,
4 | / / / pi \ \2 \ | | / / / pi \ \2 \ |
| | | pi cos| -- | 51 | | | | | | pi cos| -- | 26 | | |
| | | / pi \ / 51 pi \ \ 50 / | | | | | | / pi \ / 26 pi \ \ 25 / | | |
| sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | | | sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | |
\ \ \ \ 50 / \ 50 / 50 / / / \ \ \ \ 25 / \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 53 pi \ pi #9 53 \2 \ | | / / / 27 pi \ pi #18 27 \2 \ |
| sqrt| | #9 | ----- - 1 | - -------- | + 1 | | | sqrt| | #18 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ------------------------------------------------------------------------------------------ |, acos| ---------------------------------------------- |,
| / / / 11 pi \ \2 \ | | / / / 28 pi \ pi #17 28 \2 \ |
| | | sqrt(2) | ----- - 1 | sqrt(sqrt(5) + 5) | | | | sqrt| | #17 | ----- - 1 | - --------- | + 1 | |
| | | 11 pi sqrt(2) sqrt(sqrt(5) + 5) \ 10 / | | | \ \ \ \ 25 / 25 / / /
| sqrt| | ------------------------------- - --------------------------------------- | + 1 | |
\ \ \ 40 4 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 57 pi \ pi #8 57 \2 \ | | / / / 29 pi \ pi #16 29 \2 \ | | / / / 59 pi \ pi #7 59 \2 \ |
| sqrt| | #8 | ----- - 1 | - -------- | + 1 | | | sqrt| | #16 | ----- - 1 | - --------- | + 1 | | | sqrt| | #7 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 6 pi \ pi #20 6 \2 \ | | / / / 61 pi \ pi #6 61 \2 \ | | / / / 31 pi \ pi #15 31 \2 \ |
| sqrt| | | ---- - 1 | #20 - -------- | + 1 | | | sqrt| | #6 | ----- - 1 | - -------- | + 1 | | | sqrt| | #15 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 63 pi \ pi #5 63 \2 \ | | / / / 32 pi \ pi #14 32 \2 \ |
| sqrt| | #5 | ----- - 1 | - -------- | + 1 | | | sqrt| | #14 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ------------------------------------------------------------------------------------------ |, acos| ---------------------------------------------- |,
| / / / 13 pi \ \2 \ | | / / / 33 pi \ pi #13 33 \2 \ |
| | | sqrt(2) | ----- - 1 | sqrt(5 - sqrt(5)) | | | | sqrt| | #13 | ----- - 1 | - --------- | + 1 | |
| | | \ 10 / sqrt(2) pi sqrt(5 - sqrt(5)) 13 | | | \ \ \ \ 25 / 25 / / /
| sqrt| | --------------------------------------- - ------------------------------- | + 1 | |
\ \ \ 4 40 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 67 pi \ pi #4 67 \2 \ | | / / / 34 pi \ pi #12 34 \2 \ | | / / / 69 pi \ pi #3 69 \2 \ |
| sqrt| | #4 | ----- - 1 | - -------- | + 1 | | | sqrt| | #12 | ----- - 1 | - --------- | + 1 | | | sqrt| | #3 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 7 pi \ pi #19 7 \2 \ | | / / / 71 pi \ pi #2 71 \2 \ | | / / / 36 pi \ pi #11 36 \2 \ |
| sqrt| | | ---- - 1 | #19 - -------- | + 1 | | | sqrt| | #2 | ----- - 1 | - -------- | + 1 | | | sqrt| | #11 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, 0, acos| ---------------------------------------------- |,
| / / / 73 pi \ pi #1 73 \2 \ | | / / / 37 pi \ pi #10 37 \2 \ | | / / / 38 pi \ pi #10 38 \2 \ |
| sqrt| | #1 | ----- - 1 | - -------- | + 1 | | | sqrt| | #10 | ----- - 1 | - --------- | + 1 | | | sqrt| | #10 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 77 pi \ pi #1 77 \2 \ | | / / / 39 pi \ pi #11 39 \2 \ | | / / / 79 pi \ pi #2 79 \2 \ |
| sqrt| | #1 | ----- - 1 | - -------- | + 1 | | | sqrt| | #11 | ----- - 1 | - --------- | + 1 | | | sqrt| | #2 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 8 pi \ pi #19 8 \2 \ | | / / / 81 pi \ pi #3 81 \2 \ | | / / / 41 pi \ pi #12 41 \2 \ |
| sqrt| | | ---- - 1 | #19 - -------- | + 1 | | | sqrt| | #3 | ----- - 1 | - -------- | + 1 | | | sqrt| | #12 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 83 pi \ pi #4 83 \2 \ | | / / / 42 pi \ pi #13 42 \2 \ |
| sqrt| | #4 | ----- - 1 | - -------- | + 1 | | | sqrt| | #13 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ------------------------------------------------------------------------------------------ |, acos| ---------------------------------------------- |,
| / / / 17 pi \ \2 \ | | / / / 43 pi \ pi #14 43 \2 \ |
| | | sqrt(2) | ----- - 1 | sqrt(5 - sqrt(5)) | | | | sqrt| | #14 | ----- - 1 | - --------- | + 1 | |
| | | \ 10 / sqrt(2) pi sqrt(5 - sqrt(5)) 17 | | | \ \ \ \ 25 / 25 / / /
| sqrt| | --------------------------------------- - ------------------------------- | + 1 | |
\ \ \ 4 40 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |, acos| -------------------------------------------- |,
| / / / 87 pi \ pi #5 87 \2 \ | | / / / 44 pi \ pi #15 44 \2 \ | | / / / 89 pi \ pi #6 89 \2 \ |
| sqrt| | #5 | ----- - 1 | - -------- | + 1 | | | sqrt| | #15 | ----- - 1 | - --------- | + 1 | | | sqrt| | #6 | ----- - 1 | - -------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / / \ \ \ \ 50 / 50 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 9 pi \ pi #20 9 \2 \ | | / / / 91 pi \ pi #7 91 \2 \ | | / / / 46 pi \ pi #16 46 \2 \ |
| sqrt| | | ---- - 1 | #20 - -------- | + 1 | | | sqrt| | #7 | ----- - 1 | - -------- | + 1 | | | sqrt| | #16 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 5 / 5 / / / \ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------- |,
| / / / 93 pi \ pi #8 93 \2 \ | | / / / 47 pi \ pi #17 47 \2 \ |
| sqrt| | #8 | ----- - 1 | - -------- | + 1 | | | sqrt| | #17 | ----- - 1 | - --------- | + 1 | |
\ \ \ \ 50 / 50 / / / \ \ \ \ 25 / 25 / / /
/ 1 \ / 1 \
acos| ------------------------------------------------------------------------------------------ |, acos| ---------------------------------------------- |,
| / / / 19 pi \ \2 \ | | / / / 48 pi \ pi #18 48 \2 \ |
| | | sqrt(2) | ----- - 1 | sqrt(sqrt(5) + 5) | | | | sqrt| | #18 | ----- - 1 | - --------- | + 1 | |
| | | 19 pi sqrt(2) sqrt(sqrt(5) + 5) \ 10 / | | | \ \ \ \ 25 / 25 / / /
| sqrt| | ------------------------------- - --------------------------------------- | + 1 | |
\ \ \ 40 4 / / /
/ 1 \ / 1 \ / 1 \
acos| -------------------------------------------- |, acos| ---------------------------------------------------------- |, acos| ---------------------------------------------------------- |,
| / / / 97 pi \ pi #9 97 \2 \ | | / / / pi \ \2 \ | | / / / pi \ \2 \ |
| sqrt| | #9 | ----- - 1 | - -------- | + 1 | | | | | pi cos| -- | 49 | | | | | | pi cos| -- | 99 | | |
\ \ \ \ 50 / 50 / / / | | | / pi \ / 49 pi \ \ 25 / | | | | | | / pi \ / 99 pi \ \ 50 / | | |
| sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | | | sqrt| | cos| -- | | ----- - 1 | - --------------- | + 1 | |
\ \ \ \ 25 / \ 25 / 25 / / / \ \ \ \ 50 / \ 50 / 50 / / /
pi -- --
-- | |
4 | |
| |
| |
| |
-- --
where
/ 23 pi \
#1 == cos| ----- |
\ 50 /
/ 21 pi \
#2 == cos| ----- |
\ 50 /
/ 19 pi \
#3 == cos| ----- |
\ 50 /
/ 17 pi \
#4 == cos| ----- |
\ 50 /
/ 13 pi \
#5 == cos| ----- |
\ 50 /
/ 11 pi \
#6 == cos| ----- |
\ 50 /
/ 9 pi \
#7 == cos| ---- |
\ 50 /
/ 7 pi \
#8 == cos| ---- |
\ 50 /
/ 3 pi \
#9 == cos| ---- |
\ 50 /
/ 12 pi \
#10 == cos| ----- |
\ 25 /
/ 11 pi \
#11 == cos| ----- |
\ 25 /
/ 9 pi \
#12 == cos| ---- |
\ 25 /
/ 8 pi \
#13 == cos| ---- |
\ 25 /
/ 7 pi \
#14 == cos| ---- |
\ 25 /
/ 6 pi \
#15 == cos| ---- |
\ 25 /
/ 4 pi \
#16 == cos| ---- |
\ 25 /
/ 3 pi \
#17 == cos| ---- |
\ 25 /
/ 2 pi \
#18 == cos| ---- |
\ 25 /
sqrt(5) 1
#19 == ------- - -
4 4
sqrt(5) 1
#20 == ------- + -
4 4
vpa(q.')
ans =
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