Sin, Cos, & Tan Ratios - Louisburg USD 416 - Free Printable

Let’s solve each problem one by one. Remember:
In a right triangle, sine of an angle = (length of side opposite the angle) ÷ (length of hypotenuse).
We’ll write each answer as both a fraction and a decimal.

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1) sin A

Triangle ABC, right-angled at B.
Side opposite to angle A is BC = 24.
Hypotenuse is AC = 30.
→ sin A = 24/30 = 4/5 (simplified) → decimal: 0.8

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2) sin C

Triangle ABC, right-angled at B.
Side opposite to angle C is AB = 32.
Hypotenuse is AC = 40.
→ sin C = 32/40 = 4/5 → decimal: 0.8

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3) sin C

Triangle ABC, right-angled at B.
Side opposite to angle C is AB = 7.
Hypotenuse is AC = 25.
→ sin C = 7/25 → decimal: 0.28

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4) sin C

Triangle ABC, right-angled at B.
Side opposite to angle C is AB = 18.
Hypotenuse is AC = 30.
→ sin C = 18/30 = 3/5 → decimal: 0.6

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5) sin Y

Triangle XYZ, right-angled at Y. Wait — if it’s right-angled at Y, then angle Y is 90°, and sin(90°) = 1. But let’s check the diagram description: sides are XY=21, YZ=28, XZ=35. Since 21² + 28² = 441 + 784 = 1225 = 35², yes — right angle at Y.
So sin Y = sin(90°) = 1/1 → decimal: 1.0

But wait — maybe the question meant sin of another angle? Let me re-read: “sin Y” — and Y is the right angle. So yes, sin Y = 1.

However, sometimes problems label the right angle but ask for sine of an acute angle. Let me double-check the labeling:
It says “sin Y”, and in triangle XYZ with right angle at Y — so angle Y is 90°.
✔ Correct: sin Y = 1

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6) sin Z

Triangle XYZ, right-angled at J? Wait — labels are Z, J, X. Right angle at J.
So angle Z is at top. Side opposite to Z is JX = 30. Hypotenuse is ZX = 34.
→ sin Z = 30/34 = 15/17 → decimal ≈ 0.882 (we’ll round to 3 decimals unless specified)

Wait — let’s confirm:
Right angle at J → so legs are ZJ = 16, JX = 30; hypotenuse ZX = 34.
Angle Z is at vertex Z → opposite side is JX = 30.
Yes → sin Z = 30/34 = 15/17 ≈ 0.882

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7) sin Y

Triangle XYZ, right-angled at Y. Again, same as #5? Sides: ZY=20, YX=15, ZX=25. Check: 15²+20²=225+400=625=25² → right angle at Y.
So sin Y = sin(90°) = 1/1 → 1.0

Again, if they mean angle Y which is 90°, then yes.

But perhaps there's a mislabel? The problem says “sin Y” and Y is the right angle. We’ll go with that.

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8) sin A

Triangle ABC, right-angled at B.
Angle A is at top. Opposite side is BC = 8. Hypotenuse is AC = 17.
→ sin A = 8/17 → decimal ≈ 0.471

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9) sin A

Triangle ABC, right-angled at B.
Angle A is at left. Opposite side is BC = 48. Hypotenuse is AC = 50.
→ sin A = 48/50 = 24/25 → decimal: 0.96

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10) sin Z

Triangle XYZ, right-angled at Y.
Angle Z is at right end. Opposite side is XY = 30. Hypotenuse is XZ = 50.
→ sin Z = 30/50 = 3/5 → decimal: 0.6

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Now let’s compile all answers clearly:

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Final Answer:

1) sin A = 4/5 = 0.8
2) sin C = 4/5 = 0.8
3) sin C = 7/25 = 0.28
4) sin C = 3/5 = 0.6
5) sin Y = 1/1 = 1.0
6) sin Z = 15/17 ≈ 0.882
7) sin Y = 1/1 = 1.0
8) sin A = 8/17 ≈ 0.471
9) sin A = 24/25 = 0.96
10) sin Z = 3/5 = 0.6