Apply The Ratio

1) If tan θ = 5/12, what is the length of the opposite side?

5

13

√20

12

A tangent of 5/12 uses “opposite over adjacent,” so the opposite side is 5.

Explanation

A tangent of 5/12 uses “opposite over adjacent,” so the opposite side is 5.

2) If sin θ = 7/25, what is the length of the opposite side?

25

7

√56

24

A sine of 7/25 uses “opposite over hypotenuse,” so the opposite side is 7

Explanation

A sine of 7/25 uses “opposite over hypotenuse,” so the opposite side is 7

3) If tan θ = 9/40, what is the opposite side?

7

41

49

9

A tangent of 9/40 uses “opposite over adjacent,” so the opposite side is 9.

Explanation

A tangent of 9/40 uses “opposite over adjacent,” so the opposite side is 9.

4) If sin θ = 12/13, what is the length of the opposite side?

√20

12

13

5

A sine of 12/13 uses “opposite over hypotenuse,” so the opposite side is 12.

Explanation

A sine of 12/13 uses “opposite over hypotenuse,” so the opposite side is 12.

5) If cos θ = 15/17, what is the adjacent side length?

8

17

12

15

A cosine of 15/17 uses “adjacent over hypotenuse,” so the adjacent side is 15.

Explanation

A cosine of 15/17 uses “adjacent over hypotenuse,” so the adjacent side is 15.

6) If sin θ = 3/5, what is the hypotenuse?

√34

4

5

3

A sine of 3/5 uses “opposite over hypotenuse,” so the hypotenuse is 5.

Explanation

A sine of 3/5 uses “opposite over hypotenuse,” so the hypotenuse is 5.

7) If sin θ = 5/13, what is the adjacent side length?

25

12

13

5

A sine of 5/13 fits the 5–12–13 triangle, so the adjacent side is 12.

Explanation

A sine of 5/13 fits the 5–12–13 triangle, so the adjacent side is 12.

8) If tan θ = 15/8, what is the length of the hypotenuse?

8

15

17

23

A tangent of 15/8 fits the 8–15–17 triangle, so the hypotenuse is 17.

Explanation

A tangent of 15/8 fits the 8–15–17 triangle, so the hypotenuse is 17.

9) If cos θ = 9/41, what is the length of the opposite side?

41

40

50

√160

A cosine of 9/41 has adjacent 9 and hypotenuse 41, so the opposite side is: 40.

Explanation

A cosine of 9/41 has adjacent 9 and hypotenuse 41, so the opposite side is: 40.

10) If tan θ = 5/12 what is the length of the hypotenuse?

√119

12

13

5

A tangent of 5/12 fits the 5–12–13 triangle, so the hypotenuse is 13.

Explanation

A tangent of 5/12 fits the 5–12–13 triangle, so the hypotenuse is 13.

11) If tan θ = 12/5, what is the length of the hypotenuse?

15

13

12

5

A tangent of 12/5 uses a 5–12–13 triangle, so the hypotenuse is 13.

Explanation

A tangent of 12/5 uses a 5–12–13 triangle, so the hypotenuse is 13.

12) If cos θ = 3/5, what is the length of the hypotenuse?

√5

3

5

4

A cosine of 3/5 means “adjacent 3 over hypotenuse 5,” so the hypotenuse is 5.

Explanation

A cosine of 3/5 means “adjacent 3 over hypotenuse 5,” so the hypotenuse is 5.

13) If cos θ = 8/17, what is the length of the adjacent side?

√15

15

8

17

A cosine of 8/17 uses “adjacent over hypotenuse,” so the adjacent side is 8.

Explanation

A cosine of 8/17 uses “adjacent over hypotenuse,” so the adjacent side is 8.

14) If tan θ = 7/24, what is the adjacent side length?

√31

25

24

7

A tangent of 7/24 uses “opposite over adjacent,” so the adjacent side is 24.

Explanation

A tangent of 7/24 uses “opposite over adjacent,” so the adjacent side is 24.

15) If sin θ = 20/29, what is the adjacent side?

21

29

20

√440

A sine of 20/29 has opposite 20 and hypotenuse 29, so the adjacent side is 21.

Explanation

A sine of 20/29 has opposite 20 and hypotenuse 29, so the adjacent side is 21.

16) If cos θ = 21/29, what is the length of the opposite side?

20

8

21

29

A cosine of 21/29 has adjacent 21 and hypotenuse 29, so the opposite side is 20.

Explanation

A cosine of 21/29 has adjacent 21 and hypotenuse 29, so the opposite side is 20.

17) If sin θ = 2/5 for an acute angle θ, what is the length of the hypotenuse?

3

5

√21

2

A sine of 2/5 means “opposite 2 over hypotenuse 5,” so the hypotenuse is 5.

Explanation

A sine of 2/5 means “opposite 2 over hypotenuse 5,” so the hypotenuse is 5.

18) If cos θ = 4/5, find the opposite side length.

3

√41

4

5

A cosine of 4/5 fits the 3–4–5 triangle, so the opposite side is 3.

Explanation

A cosine of 4/5 fits the 3–4–5 triangle, so the opposite side is 3.

19) If cos θ = 5/13, what is the length of the opposite side?

5

12

13

25

A cosine of 5/13 has adjacent 5 and hypotenuse 13, so the opposite side is 12.

Explanation

A cosine of 5/13 has adjacent 5 and hypotenuse 13, so the opposite side is 12.

20) If sin θ = 40/41, what is the length of the adjacent side?

√80

40

41

9

A sine of 40/41 has opposite 40 and hypotenuse 41, so the adjacent side is 9.

Explanation

A sine of 40/41 has opposite 40 and hypotenuse 41, so the adjacent side is 9.

Apply the Ratio