Trigonometric Ratios Sohcahtoa Worksheet | PDF - Free Printable

Let’s solve each part step by step.

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Problem 1: Find the ratios for triangle ABC

We are given a right triangle with:

- Angle A at top
- Right angle at B
- Side AB = 7 (adjacent to angle A)
- Side BC = 7√3 (opposite to angle A)
- Hypotenuse AC = 14

Recall SOHCAHTOA:

- sin A = opposite / hypotenuse
- cos A = adjacent / hypotenuse
- tan A = opposite / adjacent

So:

→ sin A = BC / AC = (7√3) / 14 = √3 / 2
→ cos A = AB / AC = 7 / 14 = 1/2
→ tan A = BC / AB = (7√3) / 7 = √3

✔ These are standard values — you might recognize them from special triangles (30-60-90).

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Problem 2: Find value of each expression

These are standard trig values:

a) sin 30° = 1/2
b) cos 45° = √2 / 2 ≈ 0.707, but since it says “find the value”, and no rounding specified, we leave as exact: √2 / 2
c) tan 45° = 1

But looking at the boxes — they’re small squares. Probably expecting simplified fractions or simple decimals? But in math class, exact forms are preferred unless told otherwise.

Wait — let’s check context. Problem 3 says “round to nearest tenth”, so maybe here too? But Problem 2 doesn’t say that. So better to give exact values.

Actually, looking again — the boxes are empty, no instruction to round. So:

a) sin 30° = 0.5 or 1/2 → both acceptable, but 1/2 is more precise
b) cos 45° = √2 / 2 — but if they want decimal, it’s ~0.707 → but again, no rounding instruction
c) tan 45° = 1

Since this is likely middle/high school level, and these are memorized values, I’ll write:

a) 1/2
b) √2 / 2
c) 1

But wait — sometimes worksheets expect decimals for these. Let me think... In many US schools, they teach:

sin 30° = 0.5
cos 45° ≈ 0.707 → but often written as √2/2
tan 45° = 1

I think safest is to use exact forms unless told to approximate.

But looking at Problem 3, it explicitly says “round to nearest tenth”, implying Problems 1 and 2 may not need rounding. So I’ll go with exact.

However, in Problem 1, we got √3/2, 1/2, √3 — which are exact.

So for Problem 2:

a) sin 30° = 1/2
b) cos 45° = √2 / 2
c) tan 45° = 1

But let me double-check — actually, in some curricula, they write cos 45° as 1/√2, but √2/2 is rationalized and preferred.

Yes.

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Problem 3: Use trig ratios to find missing variables. Round to nearest tenth.

Part a:

Triangle has angles 45°, 45°, 90° → isosceles right triangle.

Hypotenuse = 16

In 45-45-90 triangle, legs are equal, and each leg = hypotenuse / √2

So:

x = y = 16 / √2

Rationalize: (16√2)/2 = 8√2

Now compute numerically: √2 ≈ 1.4142 → 8 * 1.4142 ≈ 11.3136 → round to nearest tenth: 11.3

So x = 11.3, y = 11.3

Alternatively, using trig:

For angle 45°, sin 45° = opposite/hypotenuse = x / 16

sin 45° = √2 / 2 ≈ 0.7071

So x = 16 * 0.7071 ≈ 11.3136 → 11.3

Same for y.

✔ Correct.

Part b:

Right triangle, right angle at top right.

Angle at bottom right = 60°

So angle at top left = z° = 180 - 90 - 60 = 30° → so z = 30

Side adjacent to 60° is y (vertical side), side opposite to 60° is 13 (horizontal side)

Wait — let’s label:

Vertices:

- Top right: right angle
- Bottom right: 60° angle
- Top left: z° angle

Sides:

- Horizontal side (top): 13 → this is opposite to the 60° angle? No.

Actually, from the diagram description:

It says: “b” triangle has:

- Top side labeled 13 (horizontal)
- Right side labeled y (vertical)
- Angle at bottom right is 60°
- Angle at top left is z°

Since it’s a right triangle with right angle at top right, then:

The side of length 13 is adjacent to angle z°, and opposite to angle 60°.

Angle at bottom right is 60°, so:

- Opposite side to 60° is the top side = 13
- Adjacent side to 60° is the vertical side = y
- Hypotenuse is the slanted side

So:

tan(60°) = opposite / adjacent = 13 / y

tan(60°) = √3 ≈ 1.732

So:

1.732 = 13 / y → y = 13 / 1.732 ≈ ?

Calculate: 13 ÷ 1.732 ≈ 7.505 → round to nearest tenth: 7.5

Also, z° = 30°, as calculated.

Confirm with another ratio:

sin(60°) = opposite/hypotenuse = 13 / hypotenuse

But we don’t need hypotenuse.

Or cos(60°) = adjacent/hypotenuse = y / hypotenuse

But easier with tan.

Yes.

So:

z° = 30
y = 7.5

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Now, compiling all answers:

Problem 1:

sin A = √3 / 2
cos A = 1/2
tan A = √3

Problem 2:

a) sin 30° = 1/2
b) cos 45° = √2 / 2
c) tan 45° = 1

Problem 3:

a) x = 11.3, y = 11.3
b) z° = 30, y = 7.5

But wait — in Problem 2, should I write decimals? The problem doesn’t specify, but in many worksheets, for these standard angles, they accept fractions or radicals.

However, looking back at Problem 1, they have √3 and √3/2, so radicals are fine.

But for Problem 2, part b: cos 45° = √2 / 2 — that’s fine.

Some might write it as 0.707, but since no rounding instruction, exact is better.

Similarly, sin 30° = 0.5 or 1/2 — both fine, but 1/2 is exact.

I think we’re good.

Final answers:

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Final Answer:
1. sin A = √3/2, cos A = 1/2, tan A = √3
2. a) 1/2, b) √2/2, c) 1
3. a) x = 11.3, y = 11.3; b) z° = 30, y = 7.5