10.1 Sin, Cos, and Tan - Practice Problems.ia1 - Free Printable

To solve these problems, we need to use the definitions of sine, cosine, and tangent in right triangles.

Recall:
- sin(angle) = opposite side / hypotenuse
- cos(angle) = adjacent side / hypotenuse
- tan(angle) = opposite side / adjacent side

Let’s go problem by problem.

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1) sin Z

Triangle XYZ, right angle at Y.
- Angle Z is at vertex Z.
- Opposite side to angle Z is XY = 40
- Hypotenuse is XZ = 50

So, sin Z = 40/50 = 4/5 = 0.8

✔ Check: 40 ÷ 50 = 0.8 → correct.

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2) sin Z

Triangle XYZ, right angle at Y.
- Angle Z is at vertex Z.
- Opposite side to angle Z is XY = 12
- Hypotenuse is XZ = 37

So, sin Z = 12/37

✔ Check: 12 and 37 have no common factors → fraction is simplified.

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3) cos A

Triangle ABC, right angle at B.
- Angle A is at vertex A.
- Adjacent side to angle A is AB = 30
- Hypotenuse is AC = 50

So, cos A = 30/50 = 3/5 = 0.6

✔ Check: 30 ÷ 50 = 0.6 → correct.

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4) cos C

Triangle ABC, right angle at B.
- Angle C is at vertex C.
- Adjacent side to angle C is BC = 16
- Hypotenuse is AC = 34

So, cos C = 16/34 = 8/17

✔ Check: divide numerator and denominator by 2 → 8/17 → correct.

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5) tan Z

Triangle XYZ, right angle at Y.
- Angle Z is at vertex Z.
- Opposite side to angle Z is XY = 9
- Adjacent side to angle Z is YZ = 41

Wait — let’s check the triangle.

Actually, looking again:

In triangle XYZ, right angle at Y.

Angle Z is at Z.

Opposite side to angle Z is XY = 9? But XY is labeled as 9? Wait — let me re-read.

The triangle has:

- Side XY = 9 (but that’s not opposite to Z)
Actually, if right angle is at Y, then:

Vertices: X, Y, Z with right angle at Y.

So sides:

- XY and YZ are legs
- XZ is hypotenuse = 46

Angle at Z:

- Opposite side is XY = 9
- Adjacent side is YZ = 41

Yes! So tan Z = opposite / adjacent = 9 / 41

✔ Check: 9 and 41 share no common factors → already simplified.

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6) tan A

Triangle ABC, right angle at B.

Angle A is at vertex A.

Opposite side to angle A is BC = 21

Adjacent side to angle A is AB = 25

So tan A = 21 / 25

✔ Check: 21 and 25 have no common factors → correct.

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Final Answers:

1) 0.8
2) 12/37
3) 0.6
4) 8/17
5) 9/41
6) 21/25

But since the question says “find the value”, and some are fractions, we can leave them as fractions unless decimal is requested. The original doesn’t specify, so fractions are fine — but for consistency, maybe write decimals where exact? Actually, 12/37 is repeating decimal, so better to leave as fraction.

Similarly, 8/17, 9/41, 21/25 — all are fine as fractions.

But 40/50 and 30/50 simplify nicely to decimals.

However, in math class, often they prefer simplified fractions.

Let me write all as simplified fractions or decimals as appropriate.

Actually, let's standardize:

For problems where it simplifies to a terminating decimal, we can write decimal; otherwise, fraction.

But to be safe, since the problem doesn't specify, I'll write exact values — fractions reduced.

So:

1) 40/50 = 4/5
2) 12/37
3) 30/50 = 3/5
4) 16/34 = 8/17
5) 9/41
6) 21/25

All are simplified.

Alternatively, if decimals are acceptable:

1) 0.8
2) ≈0.3243... but better to leave as fraction
Since the problem says "value", and in trig ratios, fractions are standard.

I think fractions are preferred here.

So final answers:

1) \frac{4}{5}
2) \frac{12}{37}
3) \frac{3}{5}
4) \frac{8}{17}
5) \frac{9}{41}
6) \frac{21}{25}

But the user might expect boxed answers per problem.

Since the instruction is to provide final answer after reasoning, and there are 6 problems, I should list them clearly.

Looking back at the response format, it says "Final Answer:" and then the answer. Since there are multiple parts, I’ll list them.

Perhaps:

Final Answer:
1) \frac{4}{5}
2) \frac{12}{37}
3) \frac{3}{5}
4) \frac{8}{17}
5) \frac{9}{41}
6) \frac{21}{25}

But to match the format, maybe just write the values.

Since the user said "solve the problem" and there are 6 subproblems, I'll present all.

I think this is correct.

Double-checking problem 5:

Triangle XYZ, right angle at Y.

Sides: XY = 9, YZ = 41, XZ = 46.

Angle at Z:

- Opposite side is XY = 9 (since from Z, opposite is the side not touching Z, which is XY)
- Adjacent side is YZ = 41 (side touching Z and the right angle)

Yes, tan Z = opposite/adjacent = 9/41. Correct.

Problem 6: Triangle ABC, right angle at B.

Angle at A:

- Opposite side is BC = 21
- Adjacent side is AB = 25

tan A = 21/25. Correct.

All good.

Final Answer:
1) \frac{4}{5}
2) \frac{12}{37}
3) \frac{3}{5}
4) \frac{8}{17}
5) \frac{9}{41}
6) \frac{21}{25}