Trigonometric Ratios Worksheets - Math Monks - Free Printable

Let’s solve each problem step by step. We’ll use the definitions of sine, cosine, and tangent in right triangles:

- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent

We must identify which side is opposite, adjacent, or hypotenuse relative to the angle given.

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Problem 1: sin C in triangle ABC (right angle at B)

Triangle ABC:
- Right angle at B → so AC is hypotenuse = 50
- Angle C is at vertex C
- Opposite to angle C: AB = 14
- Adjacent to angle C: BC = 48
- Hypotenuse: AC = 50

So,
sin C = opposite/hypotenuse = AB/AC = 14/50 = 7/25 = 0.28

✔ Check: 14 ÷ 50 = 0.28 — correct.

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Problem 2: sin Z in triangle XYZ (right angle at X)

Triangle XYZ:
- Right angle at X → so YZ is hypotenuse = 40
- Angle Z is at vertex Z
- Opposite to angle Z: XY = 24
- Adjacent to angle Z: XZ = 32
- Hypotenuse: YZ = 40

So,
sin Z = opposite/hypotenuse = XY/YZ = 24/40 = 3/5 = 0.6

✔ Check: 24 ÷ 40 = 0.6 — correct.

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Problem 3: tan A in triangle ABC (right angle at B)

Triangle ABC:
- Right angle at B → so AC is hypotenuse? Wait — let’s check sides:
- AB = 20
- BC = 21
- AC = 29 → this is the longest side → yes, hypotenuse
- Right angle at B → so legs are AB and BC

Angle A is at vertex A.

Opposite to angle A: BC = 21
Adjacent to angle A: AB = 20
Hypotenuse: AC = 29

So,
tan A = opposite/adjacent = BC/AB = 21/20 = 1.05

✔ Check: 21 ÷ 20 = 1.05 — correct.

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Problem 4: tan B in triangle ABC (right angle at A)

Triangle ABC:
- Right angle at A → so BC is hypotenuse = 50
- Legs: AB = 40, AC = 30

Angle B is at vertex B.

Opposite to angle B: AC = 30
Adjacent to angle B: AB = 40
Hypotenuse: BC = 50

So,
tan B = opposite/adjacent = AC/AB = 30/40 = 3/4 = 0.75

✔ Check: 30 ÷ 40 = 0.75 — correct.

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Problem 5: cos X in triangle XYZ (right angle at Y)

Triangle XYZ:
- Right angle at Y → so XZ is hypotenuse = 29
- Legs: XY = 21, YZ = 20

Angle X is at vertex X.

Adjacent to angle X: XY = 21
Opposite to angle X: YZ = 20
Hypotenuse: XZ = 29

So,
cos X = adjacent/hypotenuse = XY/XZ = 21/29 ≈ ?

Let’s calculate: 21 ÷ 29 = ?

29 × 0.7 = 20.3 → too low
29 × 0.72 = 20.88
29 × 0.724 = 20.996 → very close to 21

Actually, 21 ÷ 29 = 0.724137... → but since it says “find the value”, and no rounding specified here, we can leave as fraction or decimal? Looking at other problems, they seem to expect exact fractions or decimals if nice.

But 21/29 doesn’t simplify. Let’s see what the worksheet expects — probably decimal only for calculator ones (problems 7–8). For 1–6, likely simplified fraction or exact decimal.

Wait — 21/29 is already simplified. But maybe they want decimal? Let me check context.

Looking back: Problems 1–6 don’t say “use calculator”, while 7–8 do. So for 1–6, we should give exact values (fractions or terminating decimals).

21/29 is not a terminating decimal. Hmm.

Wait — perhaps I made a mistake?

Double-check triangle:

Points: X, Y, Z. Right angle at Y.

So sides:

XY = 21 (horizontal leg)

YZ = 20 (vertical leg)

XZ = 29 (hypotenuse) — yes, because 20² + 21² = 400 + 441 = 841 = 29² → correct.

Angle X: between hypotenuse XZ and leg XY.

So adjacent to X is XY = 21, hypotenuse is XZ = 29 → cos X = 21/29.

Since 21 and 29 share no common factors, answer is 21/29.

But let’s see if the worksheet expects decimal? Problem 6 has numbers that divide nicely.

Perhaps for consistency, we’ll write as fraction unless it terminates.

But looking at problem 1: 14/50 = 7/25 = 0.28 — they might accept either.

To be safe, let’s compute decimal to 4 places only if needed — but instruction says for 7–8 to use calculator to nearest ten-thousandth. For 1–6, probably exact form.

I think 21/29 is acceptable. But let’s hold off and see others.

Actually, let’s proceed and decide later.

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Problem 6: cos C in triangle ABC (right angle at B)

Triangle ABC:
- Right angle at B → so AC is hypotenuse = 15
- Legs: AB = 9, BC = 12

Check: 9² + 12² = 81 + 144 = 225 = 15² → correct.

Angle C is at vertex C.

Adjacent to angle C: BC = 12
Opposite to angle C: AB = 9
Hypotenuse: AC = 15

So,
cos C = adjacent/hypotenuse = BC/AC = 12/15 = 4/5 = 0.8

✔ Check: 12 ÷ 15 = 0.8 — correct.

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Now back to Problem 5: cos X = 21/29

Since 21/29 does not simplify and is not a terminating decimal, but the problem doesn’t specify format, and others are nice decimals or fractions, perhaps we should leave as fraction.

But let’s calculate it anyway for reference: 21 ÷ 29 ≈ 0.7241 — but since no calculator instruction, better to leave as fraction.

However, looking at the worksheet layout, all blanks are lines — probably expecting numerical answers. And problem 1 was 14/50=0.28, which is decimal.

In problem 1, 14/50 reduces to 7/25, but they might write 0.28.

Similarly, problem 2: 24/40=0.6

Problem 3: 21/20=1.05

Problem 4: 30/40=0.75

Problem 6: 12/15=0.8

All these are terminating decimals.

Problem 5: 21/29 — let's divide: 21 ÷ 29 = 0.724137931... — non-terminating.

But wait — is there a mistake? Let me double-check the triangle labeling.

Problem 5: Triangle XYZ, right angle at Y.

Sides: XY = 21, YZ = 20, XZ = 29.

Angle X: so from point X, the sides are:

- To Y: length 21 (this is adjacent to angle X)
- To Z: length 29 (hypotenuse)
- The side opposite angle X is YZ = 20.

Yes, so cos X = adjacent/hypotenuse = XY / XZ = 21/29.

Perhaps the worksheet expects the fractional form. Or maybe I need to write it as a decimal rounded? But instructions for 1-6 don't say to round.

Looking at the bottom part: "Use a calculator to find the value of each to the nearest ten-thousandth" only for 7 and 8. So for 1-6, exact values are expected.

21/29 is exact. But let's see if it simplifies — gcd(21,29)=1, so no.

Perhaps they want it as a decimal anyway? But that would be inconsistent.

Another thought: maybe I misidentified the angle.

Angle X — in triangle XYZ, with right angle at Y.

Vertices: X, Y, Z.

Right angle at Y, so angles at X and Z are acute.

Cosine of angle X: adjacent over hypotenuse.

Adjacent side to angle X is the leg that forms the angle with the hypotenuse — that's XY, since from X, the two sides are XY and XZ; XY is the leg, XZ is hypotenuse.

Yes.

Perhaps the answer is 21/29. I'll go with that for now.

But let's move to 7 and 8.

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Problem 7: sin 77°

Use calculator. Make sure it's in degree mode.

sin 77° ≈ ?

I know sin 90° = 1, sin 60° ≈ 0.866, sin 75° ≈ 0.9659, so sin 77° should be higher.

Calculate: sin 77° = ?

Using calculator: sin(77) = 0.974370064785...

To nearest ten-thousandth: look at fifth decimal place.

0.97437 → fourth decimal is 3, fifth is 7 ≥ 5, so round up: 0.9744

✔ sin 77° ≈ 0.9744

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Problem 8: tan 17°

tan 17° = ?

tan 0° = 0, tan 45° = 1, so tan 17° is small.

Calculator: tan(17) = 0.305730681458...

To nearest ten-thousandth: 0.30573 → fourth decimal is 7, fifth is 3 < 5, so keep as 0.3057

Wait: 0.30573 — positions:

- tenths: 3
- hundredths: 0
- thousandths: 5
- ten-thousandths: 7
- hundred-thousandths: 3

So to nearest ten-thousandth, look at hundred-thousandths digit: 3 < 5, so do not round up.

Thus, 0.3057

But let me confirm calculation.

Actually, standard value: tan 17° ≈ 0.3057 — yes.

Some calculators might show more digits, but rounding to four decimal places: 0.3057

✔ tan 17° ≈ 0.3057

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Now back to Problem 5: cos X = 21/29

Since all other answers for 1-6 are terminating decimals or simple fractions, and 21/29 is approximately 0.7241, but the problem doesn't specify rounding, I think we should leave it as a fraction.

But let's see the pattern:

Problem 1: 14/50 = 0.28

Problem 2: 24/40 = 0.6

Problem 3: 21/20 = 1.05

Problem 4: 30/40 = 0.75

Problem 6: 12/15 = 0.8

All are exact decimals.

Problem 5: 21/29 is not an exact decimal. Perhaps I made a mistake in identifying the sides.

Let me read the diagram again mentally.

Problem 5: Triangle XYZ, right angle at Y.

Labeling: X --21-- Y --20-- Z, and X to Z is 29.

Angle at X: so when we say cos X, it's the cosine of the angle at vertex X.

In right triangle, for angle at X:

- The side adjacent to X is the leg that is next to X and not the hypotenuse — that's XY = 21

- The hypotenuse is XZ = 29

- So cos X = adjacent/hypotenuse = 21/29

Perhaps the worksheet expects the decimal approximation even for 1-6? But the instruction for calculator is only for 7-8.

Maybe for 5, they want 21/29, and for others, decimals.

To resolve this, let's calculate 21/29 as a decimal to four places for consistency, but since it's not required, I'm hesitant.

Another idea: perhaps in some curricula, they leave it as fraction if not terminating.

But looking at the blank lines, probably they expect a number.

Let me check online or recall — but since I can't, I'll assume that for 1-6, if it's a nice decimal, write decimal, else fraction.

21/29 is about 0.7241, but without calculator instruction, better to write fraction.

Perhaps the answer is 21/29.

I think I'll go with the fraction for accuracy.

But let's see the final answer format.

Perhaps for Problem 5, it's 21/29, and for others, decimals.

To match the style, let's write all as decimals where possible, and for 5, calculate it.

The user didn't specify, but in educational contexts, sometimes they accept both.

But to be precise, I'll provide the exact value for 5.

Let's list all answers:

1. sin C = 14/50 = 7/25 = 0.28

2. sin Z = 24/40 = 3/5 = 0.6

3. tan A = 21/20 = 1.05

4. tan B = 30/40 = 3/4 = 0.75

5. cos X = 21/29 ≈ ? but let's keep as 21/29

6. cos C = 12/15 = 4/5 = 0.8

7. sin 77° = 0.9744

8. tan 17° = 0.3057

For 5, since 21/29 is exact, and the problem doesn't ask for decimal, I'll write it as a fraction.

But let's double-check if 21/29 can be simplified — no.

Perhaps the triangle is labeled differently.

Another thought: in problem 5, is the right angle at Y, and angle X, so perhaps they want the ratio as is.

I think it's fine.

So final answers:

1. 0.28 or 7/25 — but since 14/50 reduces, and 0.28 is exact, probably 0.28

Similarly, 2. 0.6

3. 1.05

4. 0.75

5. 21/29 — but to be consistent, perhaps calculate it.

I recall that in many worksheets, if it's not nice, they might have a typo, but here 20-21-29 is a Pythagorean triple, so it's correct.

Perhaps for cos X, they expect the decimal.

Let's calculate 21 divided by 29:

29 * 0.724 = 29*0.7=20.3, 29*0.024=0.696, total 20.996, close to 21, so 0.7241 as before.

But since no instruction, I'll put 21/29 for now.

To match the format, let's see the other answers are all decimals except possibly 5.

Problem 3 is 1.05, which is decimal.

Perhaps for 5, they want 0.7241, but that would require calculator, which is not instructed.

I think the safest is to provide the exact fraction for 5.

But let's look at the worksheet title: "Finding Trigonometric Ratios" — usually they want the ratio as a number.

Perhaps in this case, for 5, it's 21/29, and students can leave it as fraction.

I'll proceed with that.

So summarizing:

1. sin C = 14/50 = 7/25 = 0.28 → I'll write 0.28

2. sin Z = 24/40 = 0.6

3. tan A = 21/20 = 1.05

4. tan B = 30/40 = 0.75

5. cos X = 21/29

6. cos C = 12/15 = 0.8

7. sin 77° = 0.9744

8. tan 17° = 0.3057

For 5, since it's not a terminating decimal, and no calculator instruction, I'll keep it as 21/29.

But to confirm, let's see if there's another way.

Perhaps the angle is different.

Another idea: in triangle XYZ, right angle at Y, angle at X, so the adjacent side is XY=21, hypotenuse XZ=29, so cos X = 21/29.

I think it's correct.

So for the final answer, I'll list them as:

1. 0.28

2. 0.6

3. 1.05

4. 0.75

5. \frac{21}{29} or 21/29

6. 0.8

7. 0.9744

8. 0.3057

But for consistency, perhaps write all as decimals, and for 5, approximate.

Since the problem for 7-8 specifies "use calculator", for 1-6, they might expect exact values, so 21/29 is fine.

I think I'll go with that.

So Final Answer:

1. 0.28

2. 0.6

3. 1.05

4. 0.75

5. \frac{21}{29}

6. 0.8

7. 0.9744

8. 0.3057

But in the response, I need to box the final answer.

Perhaps for 5, they expect the decimal, but I'll stick with fraction.

Let's calculate 21/29 exactly: it's approximately 0.724137931, so to four decimals 0.7241, but again, not instructed.

I recall that in some systems, they leave it as fraction.

To resolve, let's assume that for 1-6, if the division is exact, write decimal, else fraction.

21/29 is not exact decimal, so fraction.

So I'll proceed.

Final Answer:

1. 0.28
2. 0.6
3. 1.05
4. 0.75
5. \frac{21}{29}
6. 0.8
7. 0.9744
8. 0.3057

But in the boxed answer, perhaps write as a list.

Since the user will fill in the blanks, I'll provide the values.

For the response, I'll write:

Final Answer:
1. 0.28
2. 0.6
3. 1.05
4. 0.75
5. \frac{21}{29}
6. 0.8
7. 0.9744
8. 0.3057

But to make it clean, perhaps for 5, write 21/29.

I think it's acceptable.

So I'll go with that.