Algebra 2/Trigonometry End Behavior Worksheet with polynomial function analysis and graph matching exercises.

Let’s solve each triangle one by one. We’ll use trigonometry (sine, cosine, tangent) and the fact that angles in a triangle add up to 180°. Since all triangles are right triangles (they have a 90° angle), we can use SOH-CAH-TOA:

- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent

We’ll round answers to the nearest tenth as instructed.

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Problem 13:

Triangle ABC, right-angled at C.
Angle A = 41°, side BC = 4 (opposite to angle A), side AC = x (adjacent to angle A).

Use tangent:
tan(41°) = opposite / adjacent = 4 / x
→ x = 4 / tan(41°)
tan(41°) ≈ 0.8693
x ≈ 4 / 0.8693 ≈ 4.6

✔ Answer: x ≈ 4.6

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Problem 14:

Right triangle at C. Angle B = 57°, side AC = 10.8 (adjacent to angle B), hypotenuse AB = x.

Use cosine:
cos(57°) = adjacent / hypotenuse = 10.8 / x
→ x = 10.8 / cos(57°)
cos(57°) ≈ 0.5446
x ≈ 10.8 / 0.5446 ≈ 19.8

✔ Answer: x ≈ 19.8

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Problem 15:

Right triangle at C. Angle B = 37°, hypotenuse AB = 10.3, side AC = x (opposite to angle B).

Use sine:
sin(37°) = opposite / hypotenuse = x / 10.3
→ x = 10.3 × sin(37°)
sin(37°) ≈ 0.6018
x ≈ 10.3 × 0.6018 ≈ 6.2

✔ Answer: x ≈ 6.2

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Problem 16:

Right triangle at C. Angle A = 47°, hypotenuse AB = 3, side AC = x (adjacent to angle A).

Use cosine:
cos(47°) = adjacent / hypotenuse = x / 3
→ x = 3 × cos(47°)
cos(47°) ≈ 0.6820
x ≈ 3 × 0.6820 ≈ 2.0

✔ Answer: x ≈ 2.0

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Now, “Solve each triangle” means find all missing sides and angles. Let’s do problems 17–24 with full solving.

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Problem 17:

Right triangle at C. Angle B = 62°, side AC = 22.6 mi (opposite to angle B). Find other sides and angle A.

First, angle A = 180° - 90° - 62° = 28°

Side BC (adjacent to angle B):
tan(62°) = opposite / adjacent = 22.6 / BC
→ BC = 22.6 / tan(62°)
tan(62°) ≈ 1.8807
BC ≈ 22.6 / 1.8807 ≈ 12.0 mi

Hypotenuse AB:
sin(62°) = opposite / hypotenuse = 22.6 / AB
→ AB = 22.6 / sin(62°)
sin(62°) ≈ 0.8829
AB ≈ 22.6 / 0.8829 ≈ 25.6 mi

✔ Answers: Angle A = 28°, BC ≈ 12.0 mi, AB ≈ 25.6 mi

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Problem 18:

Right triangle at C. Angle A = 51°, side BC = 9 in (opposite to angle A). Find other sides and angle B.

Angle B = 180° - 90° - 51° = 39°

Side AC (adjacent to angle A):
tan(51°) = opposite / adjacent = 9 / AC
→ AC = 9 / tan(51°)
tan(51°) ≈ 1.2349
AC ≈ 9 / 1.2349 ≈ 7.3 in

Hypotenuse AB:
sin(51°) = opposite / hypotenuse = 9 / AB
→ AB = 9 / sin(51°)
sin(51°) ≈ 0.7771
AB ≈ 9 / 0.7771 ≈ 11.6 in

✔ Answers: Angle B = 39°, AC ≈ 7.3 in, AB ≈ 11.6 in

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Problem 19:

Right triangle at C. Angle B = 42°, side AC = 4.5 mi (adjacent to angle B). Find other sides and angle A.

Angle A = 180° - 90° - 42° = 48°

Side BC (opposite to angle B):
tan(42°) = opposite / adjacent = BC / 4.5
→ BC = 4.5 × tan(42°)
tan(42°) ≈ 0.9004
BC ≈ 4.5 × 0.9004 ≈ 4.1 mi

Hypotenuse AB:
cos(42°) = adjacent / hypotenuse = 4.5 / AB
→ AB = 4.5 / cos(42°)
cos(42°) ≈ 0.7431
AB ≈ 4.5 / 0.7431 ≈ 6.1 mi

✔ Answers: Angle A = 48°, BC ≈ 4.1 mi, AB ≈ 6.1 mi

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Problem 20:

Right triangle at C. Angle B = 53°, side AB = 5 m (hypotenuse). Find other sides and angle A.

Angle A = 180° - 90° - 53° = 37°

Side AC (adjacent to angle B):
cos(53°) = adjacent / hypotenuse = AC / 5
→ AC = 5 × cos(53°)
cos(53°) ≈ 0.6018
AC ≈ 5 × 0.6018 ≈ 3.0 m

Side BC (opposite to angle B):
sin(53°) = opposite / hypotenuse = BC / 5
→ BC = 5 × sin(53°)
sin(53°) ≈ 0.7986
BC ≈ 5 × 0.7986 ≈ 4.0 m

✔ Answers: Angle A = 37°, AC ≈ 3.0 m, BC ≈ 4.0 m

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Problem 21:

Right triangle at C. Angle A = 28°, side BC = 29.3 mi (opposite to angle A). Find other sides and angle B.

Angle B = 180° - 90° - 28° = 62°

Side AC (adjacent to angle A):
tan(28°) = opposite / adjacent = 29.3 / AC
→ AC = 29.3 / tan(28°)
tan(28°) ≈ 0.5317
AC ≈ 29.3 / 0.5317 ≈ 55.1 mi

Hypotenuse AB:
sin(28°) = opposite / hypotenuse = 29.3 / AB
→ AB = 29.3 / sin(28°)
sin(28°) ≈ 0.4695
AB ≈ 29.3 / 0.4695 ≈ 62.4 mi

✔ Answers: Angle B = 62°, AC ≈ 55.1 mi, AB ≈ 62.4 mi

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Problem 22:

Right triangle at C. Angle A = 24°, side AB = 14 mi (hypotenuse). Find other sides and angle B.

Angle B = 180° - 90° - 24° = 66°

Side BC (opposite to angle A):
sin(24°) = opposite / hypotenuse = BC / 14
→ BC = 14 × sin(24°)
sin(24°) ≈ 0.4067
BC ≈ 14 × 0.4067 ≈ 5.7 mi

Side AC (adjacent to angle A):
cos(24°) = adjacent / hypotenuse = AC / 14
→ AC = 14 × cos(24°)
cos(24°) ≈ 0.9135
AC ≈ 14 × 0.9135 ≈ 12.8 mi

✔ Answers: Angle B = 66°, BC ≈ 5.7 mi, AC ≈ 12.8 mi

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Problem 23:

Right triangle at C. Angle A = 40°, side BC = 3 cm (opposite to angle A). Find other sides and angle B.

Angle B = 180° - 90° - 40° = 50°

Side AC (adjacent to angle A):
tan(40°) = opposite / adjacent = 3 / AC
→ AC = 3 / tan(40°)
tan(40°) ≈ 0.8391
AC ≈ 3 / 0.8391 ≈ 3.6 cm

Hypotenuse AB:
sin(40°) = opposite / hypotenuse = 3 / AB
→ AB = 3 / sin(40°)
sin(40°) ≈ 0.6428
AB ≈ 3 / 0.6428 ≈ 4.7 cm

✔ Answers: Angle B = 50°, AC ≈ 3.6 cm, AB ≈ 4.7 cm

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Problem 24:

Right triangle at C. Angle B = 28°, side AC = 6 in (adjacent to angle B). Find other sides and angle A.

Angle A = 180° - 90° - 28° = 62°

Side BC (opposite to angle B):
tan(28°) = opposite / adjacent = BC / 6
→ BC = 6 × tan(28°)
tan(28°) ≈ 0.5317
BC ≈ 6 × 0.5317 ≈ 3.2 in

Hypotenuse AB:
cos(28°) = adjacent / hypotenuse = 6 / AB
→ AB = 6 / cos(28°)
cos(28°) ≈ 0.8829
AB ≈ 6 / 0.8829 ≈ 6.8 in

✔ Answers: Angle A = 62°, BC ≈ 3.2 in, AB ≈ 6.8 in

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Final Answer:
13) x ≈ 4.6
14) x ≈ 19.8
15) x ≈ 6.2
16) x ≈ 2.0
17) Angle A = 28°, BC ≈ 12.0 mi, AB ≈ 25.6 mi
18) Angle B = 39°, AC ≈ 7.3 in, AB ≈ 11.6 in
19) Angle A = 48°, BC ≈ 4.1 mi, AB ≈ 6.1 mi
20) Angle A = 37°, AC ≈ 3.0 m, BC ≈ 4.0 m
21) Angle B = 62°, AC ≈ 55.1 mi, AB ≈ 62.4 mi
22) Angle B = 66°, BC ≈ 5.7 mi, AC ≈ 12.8 mi
23) Angle B = 50°, AC ≈ 3.6 cm, AB ≈ 4.7 cm
24) Angle A = 62°, BC ≈ 3.2 in, AB ≈ 6.8 in