Trigonometry worksheet for finding trigonometric ratios in right triangles and using a calculator for angle values.

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
- Side opposite to angle C is AB = 14
- So, sin C = opposite/hypotenuse = 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
- Side opposite to angle Z is XY = 24
- So, sin Z = opposite/hypotenuse = 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 — actually, sides are labeled: CA=29, CB=21, BA=20, right angle at B.
→ So legs are AB = 20 and BC = 21; hypotenuse is AC = 29.
Angle A is at vertex A.
Opposite to angle A is BC = 21
Adjacent to angle A is AB = 20
So tan A = opposite/adjacent = 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 is AC = 30
Adjacent to angle B is AB = 40
So tan B = opposite/adjacent = 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 is XY = 21
Hypotenuse is XZ = 29
So cos X = adjacent/hypotenuse = 21/29 ≈ ?

Let’s calculate: 21 ÷ 29 = 0.724137... → but we don’t need decimal yet unless asked. The question doesn’t specify format, but since others are fractions or decimals, let’s leave as fraction unless calculator needed later. But wait — problems 7 and 8 say “use calculator”, so maybe these first six can be exact fractions.

But looking back: Problem 5 says “cos X = ___” — no instruction to round, so we can write exact value.

21/29 is already simplified.

But let me check if they want decimal? No — only 7 and 8 say “use calculator”. So for 1–6, exact values are fine.

Wait — problem 6 also doesn’t say calculator. So we’ll keep fractions where possible.

But let’s see what’s expected. In problem 1, 14/50 reduces to 7/25. Similarly here, 21/29 is simplest form.

Actually, let’s double-check labeling:

Triangle XYZ: right angle at Y → points X, Y, Z.

Side XY = 21, YZ = 20, XZ = 29.

Angle X: between sides XY and XZ.

Adjacent to angle X: XY = 21

Opposite: YZ = 20

Hypotenuse: XZ = 29

Yes, cos X = adjacent/hypotenuse = 21/29

✔ Correct.

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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
Angle C is at vertex C.
Adjacent to angle C is BC = 12
Hypotenuse is AC = 15
So cos C = adjacent/hypotenuse = 12/15 = 4/5 = 0.8

✔ Check: 12 ÷ 15 = 0.8 — correct.

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Now Problems 7 and 8 require calculator, rounded to nearest ten-thousandth (4 decimal places).

Problem 7: sin 77°

Using calculator: sin(77) ≈ 0.974370064...

Rounded to 4 decimal places: look at fifth digit → 7, so round up fourth digit from 3 to 4? Wait:

0.97437 → digits after decimal: 9 (tenths), 7 (hundredths), 4 (thousandths), 3 (ten-thousandths), 7 (hundred-thousandths)

So ten-thousandths place is the 4th decimal: currently 3, next digit is 7 ≥ 5 → round up → 0.9744

✔ sin 77° ≈ 0.9744

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Problem 8: tan 17°

Calculator: tan(17) ≈ 0.305730681...

Digits: 0.30573...

Ten-thousandths place is 4th decimal: 7, next digit is 3 < 5 → do not round up.

So tan 17° ≈ 0.3057

✔ Confirmed.

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Final Answers:

1. sin C = 14/50 = 7/25 → but usually written as decimal or reduced fraction. Since 14/50 reduces to 7/25, and 7/25 = 0.28, either is acceptable. But in context, probably decimal or simplified fraction. Looking at other answers, some are decimals. Let's go with simplified fraction unless specified.

Wait — problem 1: 14/50 simplifies to 7/25. But 7/25 is 0.28 exactly. Either is fine. I'll write as fraction unless decimal is clearer.

But in school worksheets, often they accept both. However, since problem 6 gives 12/15 which reduces to 4/5, likely they expect simplified fractions.

But let’s check problem 2: 24/40 = 3/5 = 0.6 — again, same.

To be safe, I’ll provide simplified fractions for 1–6, and decimals for 7–8 as instructed.

But problem 3: 21/20 is improper fraction — that’s okay.

Alternatively, maybe they want decimals for all? But 7 and 8 specifically say “use calculator”, implying others may not need it.

Looking at the worksheet layout, blanks are just lines — no specification. But in math class, for exact values, fractions are preferred.

However, let’s see: in problem 1, 14/50 = 0.28 exactly — so decimal is fine.

I think for consistency, since 7 and 8 require decimals, and others can be exact, I’ll write exact values (fractions or whole numbers) for 1–6, and decimals for 7–8.

But let me verify standard practice: in trig ratios, if it’s a nice decimal, sometimes they write decimal. But 21/29 is not nice, so fraction is better.

Actually, re-examining: problem 5 is 21/29 — that’s approximately 0.7241, but since no calculator instruction, leave as fraction.

Similarly, problem 3: 21/20 = 1.05 — that’s exact decimal.

Hmm. To avoid confusion, I’ll present all answers in the most appropriate form:

- If it simplifies to a terminating decimal, use decimal.
- Otherwise, use fraction.
- For 7 and 8, use decimal as instructed.

So:

1. sin C = 14/50 = 0.28 → 0.28 (since it terminates)
2. sin Z = 24/40 = 0.6 → 0.6
3. tan A = 21/20 = 1.05 → 1.05
4. tan B = 30/40 = 0.75 → 0.75
5. cos X = 21/29 → this does not terminate, so 21/29 (but wait, is that acceptable? Maybe they expect decimal? But no calculator instruction. I think fraction is fine.)
- However, looking at the worksheet, all blanks are equal size, and 21/29 might be messy. Perhaps I should compute it? But instructions say only 7 and 8 use calculator.
- Actually, let me check: in many textbooks, for such problems, if it’s not a nice number, they leave as fraction. But 21/29 is about 0.7241, but without calculator, student might not know.
- Wait — perhaps I made a mistake? Is there a way it simplifies? 21 and 29 share no common factors (29 is prime), so 21/29 is simplest.

But let’s see problem 6: 12/15 = 0.8 — nice decimal.

Perhaps for consistency, since 1-4 and 6 give nice decimals, and 5 does not, but the problem doesn't specify, I'll write 21/29 for problem 5.

However, upon second thought, in the context of "finding trigonometric ratios" with given side lengths, and no mention of calculator for 1-6, the expected answer is likely the ratio as a fraction or simplified form.

But to match the style, let's see: in problem 1, 14/50 could be left as is, but usually simplified.

I recall that in such worksheets, they often expect the ratio in simplest form, whether fraction or decimal.

For problem 5, 21/29 is already simplified, so I'll keep it as fraction.

But let's confirm with calculation: 21 divided by 29 is approximately 0.7241, but since no calculator, fraction is better.

Actually, I think I should provide the exact value as per definition.

Final decision:

- Problems 1-6: exact values (simplified fractions or decimals if terminating)
- Problems 7-8: decimals to 4 places

So:

1. sin C = 14/50 = 7/25 or 0.28 — I'll use 0.28 since it's exact and simple.
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 — leave as fraction
6. cos C = 12/15 = 4/5 = 0.8
7. sin 77° = 0.9744
8. tan 17° = 0.3057

But for problem 5, if I write 21/29, is that acceptable? Yes.

To be thorough, let's list all:

After careful consideration, I'll present:

1. 0.28
2. 0.6
3. 1.05
4. 0.75
5. \frac{21}{29} (but since the blank is for a number, perhaps decimal? But no calculator. I think fraction is fine.)

Wait — looking back at the user's image description, it's a worksheet with blanks, and for 7 and 8, it specifies "use calculator", implying for 1-6, you can do it without, so fractions are expected for non-terminating.

But 21/29 is not nice, but it's correct.

Perhaps I can write it as a decimal approximation? But that would be inconsistent.

Another idea: maybe I miscalculated which sides are which.

In problem 5: triangle XYZ, right angle at Y.

Points: X, Y, Z.

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

Angle at X: so from X, the sides are to Y and to Z.

The side adjacent to angle X is XY = 21 (since it's next to X and not the hypotenuse).

The hypotenuse is XZ = 29.

Yes, cos X = adjacent/hypotenuse = 21/29.

Correct.

I think it's fine.

For the final answer, I'll write:

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 since the output should be plain text, for problem 5, I'll write "21/29".

Similarly, others as decimals or fractions.

To make it uniform, perhaps convert all to decimals where possible, but for 5, it's irrational? No, it's rational but repeating? 21/29 = 0.724137931... repeating, so not terminating.

So best to leave as fraction.

Final Answer:

1. 0.28
2. 0.6
3. 1.05
4. 0.75
5. 21/29
6. 0.8
7. 0.9744
8. 0.3057

But let's write them in order as per the problems.

──────────────────────────────────────

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