Kuta Software - Infinite Algebra 2 - Studocu - Free Printable

Let’s solve each problem step by step. We’ll use trigonometry (sine, cosine, tangent) and the Pythagorean theorem where needed. Remember:

- In a right triangle:
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
- Pythagorean theorem: a² + b² = c² (where c is the hypotenuse)

We’ll go one problem at a time.

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Problem 1:
Given: angle = 37°, adjacent side = 8, find x (opposite), y (hypotenuse)

Step 1: Find x using tan(37°)
tan(37°) = x / 8 → x = 8 * tan(37°)
tan(37°) ≈ 0.7536 → x ≈ 8 * 0.7536 ≈ 6.029 → x ≈ 6.0

Step 2: Find y using cos(37°)
cos(37°) = 8 / y → y = 8 / cos(37°)
cos(37°) ≈ 0.7986 → y ≈ 8 / 0.7986 ≈ 10.018 → y ≈ 10.0

✔ Answer for #1: x ≈ 6.0, y ≈ 10.0

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Problem 2:
Given: angle = 41°, opposite = 10, find x (adjacent), y (hypotenuse)

Step 1: Find x using tan(41°)
tan(41°) = 10 / x → x = 10 / tan(41°)
tan(41°) ≈ 0.8693 → x ≈ 10 / 0.8693 ≈ 11.504 → x ≈ 11.5

Step 2: Find y using sin(41°)
sin(41°) = 10 / y → y = 10 / sin(41°)
sin(41°) ≈ 0.6561 → y ≈ 10 / 0.6561 ≈ 15.242 → y ≈ 15.2

✔ Answer for #2: x ≈ 11.5, y ≈ 15.2

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Problem 3:
Given: two legs = 5 and 7, find angles A and B, and hypotenuse c

Step 1: Find hypotenuse c using Pythagoras
c = √(5² + 7²) = √(25 + 49) = √74 ≈ 8.602 → c ≈ 8.6

Step 2: Find angle A (opposite to side 5)
tan(A) = 5/7 → A = arctan(5/7) ≈ arctan(0.7143) ≈ 35.5° → A ≈ 35.5°

Step 3: Find angle B (opposite to side 7)
Since it’s a right triangle, A + B = 90° → B = 90 - 35.5 = 54.5° → B ≈ 54.5°

✔ Answer for #3: A ≈ 35.5°, B ≈ 54.5°, c ≈ 8.6

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Problem 4:
Given: two legs = 6 and 8, find angles A and B, and hypotenuse c

Step 1: Hypotenuse c = √(6² + 8²) = √(36+64) = √100 = 10 → c = 10.0

Step 2: Angle A (opposite 6): tan(A) = 6/8 = 0.75 → A = arctan(0.75) ≈ 36.87° → A ≈ 36.9°

Step 3: Angle B = 90 - 36.9 = 53.1° → B ≈ 53.1°

✔ Answer for #4: A ≈ 36.9°, B ≈ 53.1°, c = 10.0

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Problem 5:
Given: angle = 28°, hypotenuse = 12, find x (adjacent), y (opposite)

Step 1: x = adjacent = 12 * cos(28°)
cos(28°) ≈ 0.8829 → x ≈ 12 * 0.8829 ≈ 10.595 → x ≈ 10.6

Step 2: y = opposite = 12 * sin(28°)
sin(28°) ≈ 0.4695 → y ≈ 12 * 0.4695 ≈ 5.634 → y ≈ 5.6

✔ Answer for #5: x ≈ 10.6, y ≈ 5.6

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Problem 6:
Given: angle = 43°, adjacent = 9, find x (opposite), y (hypotenuse)

Step 1: x = 9 * tan(43°)
tan(43°) ≈ 0.9325 → x ≈ 9 * 0.9325 ≈ 8.393 → x ≈ 8.4

Step 2: y = 9 / cos(43°)
cos(43°) ≈ 0.7314 → y ≈ 9 / 0.7314 ≈ 12.305 → y ≈ 12.3

✔ Answer for #6: x ≈ 8.4, y ≈ 12.3

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Problem 7:
Given: angle = 52°, opposite = 11, find x (adjacent), y (hypotenuse)

Step 1: x = 11 / tan(52°)
tan(52°) ≈ 1.2799 → x ≈ 11 / 1.2799 ≈ 8.594 → x ≈ 8.6

Step 2: y = 11 / sin(52°)
sin(52°) ≈ 0.7880 → y ≈ 11 / 0.7880 ≈ 13.959 → y ≈ 14.0

✔ Answer for #7: x ≈ 8.6, y ≈ 14.0

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Problem 8:
Given: angle = 35°, hypotenuse = 15, find x (adjacent), y (opposite)

Step 1: x = 15 * cos(35°)
cos(35°) ≈ 0.8192 → x ≈ 15 * 0.8192 ≈ 12.288 → x ≈ 12.3

Step 2: y = 15 * sin(35°)
sin(35°) ≈ 0.5736 → y ≈ 15 * 0.5736 ≈ 8.604 → y ≈ 8.6

✔ Answer for #8: x ≈ 12.3, y ≈ 8.6

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Problem 9:
Given: legs = 3 and 4, find angles A and B, hypotenuse c

Step 1: c = √(3² + 4²) = √(9+16) = √25 = 5 → c = 5.0

Step 2: Angle A (opposite 3): tan(A) = 3/4 = 0.75 → A = arctan(0.75) ≈ 36.87° → A ≈ 36.9°

Step 3: Angle B = 90 - 36.9 = 53.1° → B ≈ 53.1°

✔ Answer for #9: A ≈ 36.9°, B ≈ 53.1°, c = 5.0

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Problem 10:
Given: legs = 5 and 12, find angles A and B, hypotenuse c

Step 1: c = √(5² + 12²) = √(25+144) = √169 = 13 → c = 13.0

Step 2: Angle A (opposite 5): tan(A) = 5/12 ≈ 0.4167 → A = arctan(0.4167) ≈ 22.62° → A ≈ 22.6°

Step 3: Angle B = 90 - 22.6 = 67.4° → B ≈ 67.4°

✔ Answer for #10: A ≈ 22.6°, B ≈ 67.4°, c = 13.0

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Problem 11:
Given: legs = 7 and 24, find angles A and B, hypotenuse c

Step 1: c = √(7² + 24²) = √(49 + 576) = √625 = 25 → c = 25.0

Step 2: Angle A (opposite 7): tan(A) = 7/24 ≈ 0.2917 → A = arctan(0.2917) ≈ 16.26° → A ≈ 16.3°

Step 3: Angle B = 90 - 16.3 = 73.7° → B ≈ 73.7°

✔ Answer for #11: A ≈ 16.3°, B ≈ 73.7°, c = 25.0

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Problem 12:
Given: legs = 9 and 40, find angles A and B, hypotenuse c

Step 1: c = √(9² + 40²) = √(81 + 1600) = √1681 = 41 → c = 41.0

Step 2: Angle A (opposite 9): tan(A) = 9/40 = 0.225 → A = arctan(0.225) ≈ 12.68° → A ≈ 12.7°

Step 3: Angle B = 90 - 12.7 = 77.3° → B ≈ 77.3°

✔ Answer for #12: A ≈ 12.7°, B ≈ 77.3°, c = 41.0

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Final Answer:
1. x ≈ 6.0, y ≈ 10.0
2. x ≈ 11.5, y ≈ 15.2
3. A ≈ 35.5°, B ≈ 54.5°, c ≈ 8.6
4. A ≈ 36.9°, B ≈ 53.1°, c = 10.0
5. x ≈ 10.6, y ≈ 5.6
6. x ≈ 8.4, y ≈ 12.3
7. x ≈ 8.6, y ≈ 14.0
8. x ≈ 12.3, y ≈ 8.6
9. A ≈ 36.9°, B ≈ 53.1°, c = 5.0
10. A ≈ 22.6°, B ≈ 67.4°, c = 13.0
11. A ≈ 16.3°, B ≈ 73.7°, c = 25.0
12. A ≈ 12.7°, B ≈ 77.3°, c = 41.0