Question

write the expression of either sine cosine or tangent of a single angle
cos(pi/5)cos(pi/7)+sin(pi/5)sin(pi/7)

Answer 1

do you recall this identity:
\[\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b)?\]

Answer 2

do you see how you can use that identity here?

Answer 3

kind of yes

Answer 4

is it obvious to you what a and b is?
if so just replace a and b on the left hand side

Answer 5

5 and 7 correct?

Answer 6

i don't see 5 and 7

Answer 7

i see the following numbers pi/5 and pi/7

Answer 8

yes thats what i meant!

Answer 9

so it doesn't really matter what you put as a and what you put as b since cos is an even function
cos(a-b)=cos(-(a-b)) by the fact cosine is even
=cos(b-a)

Answer 10

when I plugged in a and b i got 1

Answer 11

so pi/5-pi/7=1?

I don't think so unless I have forgotten how to add/subtract fractions

Answer 12

you need to find a common denominator

Answer 13

what is the lcm (least common multiple) of 5 and 7?

Answer 14

35

Answer 15

ok
we have
\[\frac{\pi}{5}-\frac{\pi}{7}\]
and we need the same denominator which we determined that will be 35
so we have
\[\frac{7 \pi}{5 \cdot 7}-\frac{5 \pi}{7 \cdot 5}\]
first fraction we multiplied by 7/7
second fraction we multiplied by 5/5

Answer 16

now what is 7pi-5pi?

Answer 17

2pi

Answer 18

so pi/5-pi/7 is 2pi/35

Answer 19

so \[\cos(\frac{\pi}{5})\cos(\frac{\pi}{7})+\sin(\frac{\pi}{5})\sin(\frac{\pi}{7})=\cos(\frac{\pi}{5}-\frac{\pi}{7})=\cos(\frac{2\pi}{35})\]

Answer 20

thank you so so much for all of your help!

Answer 21

np