cosine/tangent help?

Question

In triangle DFG, angle G is a right angle and tan angle D = 5/7. What is the cos of angle F to the nearest thousandths?

In triangle RST, angle T is a right angle and cos of angle R =4/9. To the nearest thousandth, find the cos of angle S?

Answer 1

tanD=(opposite side)/(adjacent side)
tanD=5/7=FG/DG:
opposite side=5=FG
adjacent side=7=DG
use pythagorean theorem to find the hypotenuse.
sqrt(5^2+7^2)
=sqrt(25+49)
=8.6023...
cosF=adjacent/hypotenuse:
cosF=FG/hypotenuse
cosF=5/8.6023...
cosF=0.58

cosR=4/9
adjacent=RT=4
hypotenuse=RS=9
use pytagorean theorem again:(c^2=a^2+b^2)
9^2=4^2+b^2
81=16+b^2
65=b^2
b=sqrt65=ST=adjacent of angle S
cosS=sqrt65/9
cosS=0.895806413...
cosS=0.9

Answer 2

You need to know what SOH CAH TOA means to answer these questions.

1. Tan angle D = 5/7 = Opposite/Adjacent
So we have a right triangle with legs of lengths 5 and 7.
Using pythagorean theorem, we can find the length of the
hypotenuse, which should come out to the the square root
of 74.
Then cos F = Adjacent/Hypotenuse = 5/ sq. root of 74
Then round this answer to the nearest thousandth.

2. Similar process as # 1.

If you have more questions about this let me know

Answer 3

Question 1
FD² = 49 + 25
FD² = 74
FD = √74
cos F = 5 / √74
cos F = 0.581

Question 2
ST² = 9² - 4²
ST² = 65
ST = √65
cos S = √65 / 9
cos S = 0.896

Answer 4

cos F = 5/8.6023 = .581
cos S = 8.0623/9 = .896

Answer 5

the only side you're missing is the hypothenus. it's 8.6. so from that COS f is adj/hyp=== 5/8.6== .581