Trigonometric Ratios Worksheets - Math Monks - Free Printable

Let's solve each problem step by step using trigonometric ratios (sine, cosine, tangent) in right triangles. We'll use:

- $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
- $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
- $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

We'll also use inverse trig functions ($\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$) for finding angles.

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Problems 1–8: Find Missing Sides

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#### Problem 1
Triangle ABC:
- Right angle at B
- Angle A = 70°
- BC = 7.2 (adjacent to 70°)
- AB = y (opposite to 70°)
- AC = x (hypotenuse)

Use:
- $\tan(70^\circ) = \frac{y}{7.2} \Rightarrow y = 7.2 \cdot \tan(70^\circ)$
- $\cos(70^\circ) = \frac{7.2}{x} \Rightarrow x = \frac{7.2}{\cos(70^\circ)}$

Calculate:
- $\tan(70^\circ) \approx 2.7475$ → $y = 7.2 \cdot 2.7475 \approx 19.78$
- $\cos(70^\circ) \approx 0.3420$ → $x = \frac{7.2}{0.3420} \approx 21.05$

✔ Answer:
$x \approx 21.05$, $y \approx 19.78$

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#### Problem 2
Triangle PQR:
- Right angle at R
- Angle Q = 16°
- PQ = 20 (hypotenuse)
- PR = y (opposite to 16°)
- QR = x (adjacent to 16°)

Use:
- $\sin(16^\circ) = \frac{y}{20} \Rightarrow y = 20 \cdot \sin(16^\circ)$
- $\cos(16^\circ) = \frac{x}{20} \Rightarrow x = 20 \cdot \cos(16^\circ)$

Calculate:
- $\sin(16^\circ) \approx 0.2756$ → $y \approx 20 \cdot 0.2756 = 5.51$
- $\cos(16^\circ) \approx 0.9613$ → $x \approx 20 \cdot 0.9613 = 19.23$

✔ Answer:
$x \approx 19.23$, $y \approx 5.51$

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#### Problem 3
Right triangle:
- Angle = 50° at bottom left
- Adjacent side = 5
- Opposite side = y
- Hypotenuse = x

So:
- $\tan(50^\circ) = \frac{y}{5} \Rightarrow y = 5 \cdot \tan(50^\circ)$
- $\cos(50^\circ) = \frac{5}{x} \Rightarrow x = \frac{5}{\cos(50^\circ)}$

Calculate:
- $\tan(50^\circ) \approx 1.1918$ → $y \approx 5 \cdot 1.1918 = 5.96$
- $\cos(50^\circ) \approx 0.6428$ → $x \approx \frac{5}{0.6428} \approx 7.78$

✔ Answer:
$x \approx 7.78$, $y \approx 5.96$

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#### Problem 4
Triangle:
- Right angle at top
- One angle = 64° (bottom right)
- Side adjacent to 64° = 10 (horizontal leg)
- Opposite = x (vertical leg)
- Hypotenuse = y

So:
- $\tan(64^\circ) = \frac{x}{10} \Rightarrow x = 10 \cdot \tan(64^\circ)$
- $\cos(64^\circ) = \frac{10}{y} \Rightarrow y = \frac{10}{\cos(64^\circ)}$

Calculate:
- $\tan(64^\circ) \approx 2.0503$ → $x \approx 10 \cdot 2.0503 = 20.50$
- $\cos(64^\circ) \approx 0.4384$ → $y \approx \frac{10}{0.4384} \approx 22.81$

✔ Answer:
$x \approx 20.50$, $y \approx 22.81$

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#### Problem 5
Isosceles right triangle:
- Angles: 45°, 45°, 90°
- Legs: x and y
- Since it’s 45-45-90, legs are equal → $x = y$

But wait — which is which? Let’s assume the vertical leg is $x$, horizontal is $y$. But both are legs.

Since it’s a 45°-45°-90° triangle, the legs are equal, and hypotenuse = $x\sqrt{2}$

But we don’t have any side labeled. Wait — actually, the diagram shows a right triangle with two 45° angles, so it's isosceles.

But no side is labeled numerically? Wait — looking again, it seems like no side is labeled?

Wait — rechecking: No number is given. That can't be. Is there a typo?

Wait — actually, the image shows no numbers on this triangle? Hmm...

Wait — maybe I misread. Looking carefully: Problem 5 has a right triangle with two 45° angles, but no side labeled. That would make it unsolvable unless one side is implied.

Wait — perhaps the diagram is missing labels? Or maybe the hypotenuse or a leg is meant to be inferred?

Actually, upon closer inspection: There is no number on triangle 5. So unless we're missing something, this might be an error.

Wait — let me double-check your image description. You said "I uploaded an image" — but since I can't see it, I'm relying on your text.

But based on what you wrote: "5" has a triangle with 45°, 45°, 90°, and variables x and y, but no numerical side.

That suggests either:
- It's a general case (but then we can't find values)
- Or there's a missing label

Wait — perhaps the hypotenuse is labeled? But in your text, it's not.

Alternatively, maybe it's supposed to be same as others, but clearly, without a known side, we can't compute x and y.

Wait — unless the triangle is drawn with one side implied, but from your description, no.

Wait — perhaps in the original image, there's a number? For example, maybe the hypotenuse is 1 or something?

But in your text, it says: "5" has only angles and x, y.

So unless there's a mistake, this problem cannot be solved without more info.

But let's assume that one of the legs is 1 or something? No — better to flag this.

Wait — actually, re-examining: Maybe the triangle is meant to be solved using ratios, but since no side is given, it must be that the problem expects symbolic answer? But the others ask for numbers.

Alternatively, perhaps the diagram includes a side, but it's not typed in your message.

Given that all other problems have numbers, likely Problem 5 has a missing value.

But let's suppose the hypotenuse is 1? Then $x = y = \frac{1}{\sqrt{2}} \approx 0.707$

But that’s arbitrary.

Alternatively, maybe the vertical leg is labeled, but not shown in text.

Wait — looking at your text: “5” has “x” on vertical leg, “y” on base, and 45° at bottom right.

No number.

This is problematic.

Wait — maybe you missed typing a number? Because otherwise, it's unsolvable.

But let’s move on and come back.

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#### Problem 6
Triangle:
- Right angle at bottom right
- Angle at top = 60°
- Angle at bottom left = 30°
- Hypotenuse = 1
- Vertical leg = x (opposite 30°)
- Horizontal leg = y (adjacent to 30°)

So:
- $\sin(30^\circ) = \frac{x}{1} \Rightarrow x = \sin(30^\circ) = 0.5$
- $\cos(30^\circ) = \frac{y}{1} \Rightarrow y = \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx 0.866$

✔ Answer:
$x = 0.5$, $y \approx 0.866$

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#### Problem 7
Triangle:
- Right angle at bottom left
- Bottom angle = 30°
- Top angle = 60°
- Base = 27 (adjacent to 30°)
- Vertical leg = x (opposite 30°)
- Hypotenuse = y

So:
- $\tan(30^\circ) = \frac{x}{27} \Rightarrow x = 27 \cdot \tan(30^\circ)$
- $\cos(30^\circ) = \frac{27}{y} \Rightarrow y = \frac{27}{\cos(30^\circ)}$

Calculate:
- $\tan(30^\circ) \approx 0.5774$ → $x \approx 27 \cdot 0.5774 = 15.59$
- $\cos(30^\circ) \approx 0.8660$ → $y \approx \frac{27}{0.8660} \approx 31.18$

✔ Answer:
$x \approx 15.59$, $y \approx 31.18$

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#### Problem 8
Triangle:
- Right angle at bottom right
- Angle at top = 30°
- Angle at bottom left = 60°
- Vertical leg = 6 (opposite 30°)
- Horizontal leg = x (adjacent to 30°)
- Hypotenuse = y

So:
- $\tan(30^\circ) = \frac{6}{x} \Rightarrow x = \frac{6}{\tan(30^\circ)}$
- $\sin(30^\circ) = \frac{6}{y} \Rightarrow y = \frac{6}{\sin(30^\circ)}$

Calculate:
- $\tan(30^\circ) \approx 0.5774$ → $x \approx \frac{6}{0.5774} \approx 10.39$
- $\sin(30^\circ) = 0.5$ → $y = \frac{6}{0.5} = 12$

✔ Answer:
$x \approx 10.39$, $y = 12$

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Now back to Problem 5:
If no side is given, we cannot compute numerical values. But if it's a 45-45-90 triangle, and one leg is x, other is y, then $x = y$, and hypotenuse = $x\sqrt{2}$

But without a known side, we can't find numbers.

Wait — perhaps in the original image, the hypotenuse is labeled? Or maybe the leg is 1?

Alternatively, maybe the triangle is meant to be similar to others, but still.

Wait — perhaps it's a trick question — since both angles are 45°, the legs are equal → $x = y$

But we need a value.

Unless the diagram shows a side, but you didn't include it.

Given the context, perhaps Problem 5 is missing data.

But let’s assume that the hypotenuse is 1? Then:
- $x = y = \frac{1}{\sqrt{2}} \approx 0.707$

But that’s speculative.

Alternatively, maybe the vertical leg is labeled 'x', and base is 'y', and it's just asking for relationship? But the others ask for numbers.

Wait — perhaps in the original image, the hypotenuse is labeled, say, 10? But not specified.

Without more info, we skip or note issue.

But let's look at Problem 9 and 10 — they have sides.

Wait — maybe Problem 5 is intended to be solved symbolically, but the format asks for numbers.

Alternatively, perhaps the triangle is drawn with one leg = 1, but not stated.

Given ambiguity, I’ll assume that Problem 5 is incomplete.

But let’s proceed to Problems 9 and 10.

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Problems 9–10: Find Unknown Angles

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#### Problem 9
Right triangle:
- Right angle at bottom left
- Vertical leg = 6
- Base = 4
- Angle at bottom right = y
- Angle at top = x

We need to find angles x and y.

Use tangent:
- $\tan(y) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{4} = 1.5$
- $y = \tan^{-1}(1.5) \approx 56.31^\circ \Rightarrow \boxed{56^\circ}$ (rounded)

Then $x = 90^\circ - y = 90 - 56.31 = 33.69^\circ \Rightarrow \boxed{34^\circ}$

✔ Answer:
$x = 34^\circ$, $y = 56^\circ$

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#### Problem 10
Right triangle:
- Right angle at bottom right
- Hypotenuse = 14
- Opposite side to angle x = 7
- Adjacent = ?
- Angle at top = x
- Angle at bottom left = y

So:
- $\sin(x) = \frac{7}{14} = 0.5 \Rightarrow x = \sin^{-1}(0.5) = 30^\circ$
- Then $y = 90^\circ - 30^\circ = 60^\circ$

✔ Answer:
$x = 30^\circ$, $y = 60^\circ$

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Now back to Problem 5 — since no side is labeled, but it's a 45-45-90 triangle, perhaps the intended answer is that x = y, but we can't give numbers.

But maybe in the image, one side is labeled? For example, if the hypotenuse is 1, then $x = y = \frac{\sqrt{2}}{2} \approx 0.707$

But without confirmation, we can't assume.

Alternatively, perhaps Problem 5 is a typo, and one side is meant to be labeled.

But let’s suppose the leg is 1 — then $x = 1$, $y = 1$

But that’s arbitrary.

Alternatively, maybe the hypotenuse is 1 → $x = y = \frac{1}{\sqrt{2}} \approx 0.707$

But again, no basis.

Wait — perhaps Problem 5 has a side labeled in the image, but not in your text.

Given that, and since all others are solvable, I suspect Problem 5 is missing a number.

But let’s check if any side is implied — no.

So I’ll note that Problem 5 requires a side length to solve.

But let’s try to reconstruct.

Wait — perhaps the vertical leg is labeled 'x', and base is 'y', and it's just asking for ratio? But the instruction is to find missing sides.

Alternatively, maybe the hypotenuse is 1, and it's standard.

But since it's not stated, I’ll leave it.

But let’s assume the hypotenuse is 1 — common in such problems.

Then:
- $x = \sin(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707$
- $y = \cos(45^\circ) = \frac{\sqrt{2}}{2} \approx 0.707$

So $x = y \approx 0.707$

✔ Assuming hypotenuse = 1, then $x = y \approx 0.707$

But this is assumption-based.

Alternatively, if a leg is 1, then $x = 1$, $y = 1$

But without info, best to say: Insufficient data.

But given the pattern, likely one leg is 1, or hypotenuse is 1.

But let’s compare to Problem 6: hypotenuse = 1 → so maybe here too.

So I’ll go with:

✔ Problem 5 (assuming hypotenuse = 1):
$x \approx 0.707$, $y \approx 0.707$

But note: This is based on assumption.

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✔ Final Answers Summary:

| Problem | x | y |
|--------|------------------|------------------|
| 1 | ≈ 21.05 | ≈ 19.78 |
| 2 | ≈ 19.23 | ≈ 5.51 |
| 3 | ≈ 7.78 | ≈ 5.96 |
| 4 | ≈ 20.50 | ≈ 22.81 |
| 5 | ≈ 0.707 (assumed)| ≈ 0.707 (assumed)|
| 6 | 0.5 | ≈ 0.866 |
| 7 | ≈ 15.59 | ≈ 31.18 |
| 8 | ≈ 10.39 | 12 |
| 9 | 34° | 56° |
| 10 | 30° | 60° |

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⚠️ Note:

- Problem 5 lacks a given side length. The solution assumes the hypotenuse is 1. If a different side is given in the image, adjust accordingly.
- All other answers are calculated using trigonometric ratios and rounded to two decimal places (except angles, rounded to nearest degree).

Let me know if you can confirm the missing side in Problem 5!