Right Triangle Trig Evaluating Ratios - Free Printable

I'll solve this worksheet on evaluating trigonometric ratios for right triangles. I need to find the value of each trig function indicated for the 12 problems shown.

Let me work through each problem one by one, using the definitions of trigonometric functions in right triangles:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent
- csc(θ) = hypotenuse/opposite
- sec(θ) = hypotenuse/adjacent
- cot(θ) = adjacent/opposite

I'll start with problem 1 and work my way through all 12.

Problem 1: sec θ
In this right triangle, θ is at the bottom left. The adjacent side to θ is 8, and the hypotenuse is 17.
sec(θ) = hypotenuse/adjacent = 17/8

Problem 2: sec θ
Here, θ is at the top. The adjacent side to θ is 12, and the hypotenuse is 13.
sec(θ) = hypotenuse/adjacent = 13/12

Problem 3: cot θ
θ is at the top. The adjacent side to θ is 4, and the opposite side is 3.
cot(θ) = adjacent/opposite = 4/3

Problem 4: csc θ
θ is at the bottom left. The opposite side to θ is 8, and the hypotenuse is 17.
csc(θ) = hypotenuse/opposite = 17/8

Problem 5: csc θ
θ is at the bottom left. The opposite side to θ is 24, and the hypotenuse is 10√5.
csc(θ) = hypotenuse/opposite = (10√5)/24 = (5√5)/12

Problem 6: cos θ
θ is at the bottom right. The adjacent side to θ is 21, and the hypotenuse is 7√10.
cos(θ) = adjacent/hypotenuse = 21/(7√10) = 3/√10 = (3√10)/10

Problem 7: cot θ
θ is at the bottom left. The adjacent side to θ is 20, and the opposite side is 15.
cot(θ) = adjacent/opposite = 20/15 = 4/3

Problem 8: tan θ
θ is at the bottom left. The opposite side to θ is 22, and the adjacent side is 2√23.
tan(θ) = opposite/adjacent = 22/(2√23) = 11/√23 = (11√23)/23

Problem 9: tan θ
θ is at the bottom left. The opposite side to θ is 6, and the adjacent side is 8.
tan(θ) = opposite/adjacent = 6/8 = 3/4

Problem 10: cot θ
θ is at the bottom left. The adjacent side to θ is 4, and the opposite side is 6.
cot(θ) = adjacent/opposite = 4/6 = 2/3

Problem 11: tan θ
θ is at the top. The opposite side to θ is 3, and the adjacent side is 4.
tan(θ) = opposite/adjacent = 3/4

Problem 12: cot θ
θ is at the bottom left. The adjacent side to θ is 12, and the opposite side is 2√5.
cot(θ) = adjacent/opposite = 12/(2√5) = 6/√5 = (6√5)/5

Now I'll summarize all the answers:

1) sec θ = 17/8
2) sec θ = 13/12
3) cot θ = 4/3
4) csc θ = 17/8
5) csc θ = (5√5)/12
6) cos θ = (3√10)/10
7) cot θ = 4/3
8) tan θ = (11√23)/23
9) tan θ = 3/4
10) cot θ = 2/3
11) tan θ = 3/4
12) cot θ = (6√5)/5