Quiz & Worksheet - Inverse Trigonometric Function Problems

Let’s solve each problem one by one. These are right triangles, and we’re being asked to find an angle labeled “x”. Since we know two sides of the triangle, we can use inverse trigonometric functions (like sin⁻¹, cos⁻¹, or tan⁻¹) to find the angle.

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Problem 1:

We have a right triangle with:
- Hypotenuse = 7
- Adjacent side to angle x = 3

Since we have adjacent and hypotenuse, we use cosine:

> cos(x) = adjacent / hypotenuse = 3/7

Now take the inverse cosine:

> x = cos⁻¹(3/7)

Let’s calculate that:

3 ÷ 7 ≈ 0.4286
cos⁻¹(0.4286) ≈ ?

Using a calculator (make sure it’s in degree mode):

cos⁻¹(0.4286) ≈ 64.6 degrees → rounds to about 65 degrees

Looking at the options: 65, 25, 23, 99 → 65 is there

✔ So answer for #1 is 65

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Problem 2:

Right triangle with:
- Hypotenuse = 25
- Opposite side to angle x = 11

Wait — let’s check the diagram again. The right angle is at the top left. Side labeled “x” is vertical on the left. Side labeled “11” is horizontal on top. Hypotenuse is 25.

So angle x is at the bottom left corner.

That means:
- For angle x:
- Opposite side = 11 (top side)
- Adjacent side = x (left side) ← but we don’t know x yet!
- Hypotenuse = 25

Actually — wait! We’re solving for angle x, not side x. In this case, “x” is the angle, not a side length. Looking back at the image description — yes, in all three problems, “x” is the angle we’re solving for.

In Problem 2:
Angle x is at the bottom left.
Opposite side to x = 11 (the top side)
Hypotenuse = 25

So:

> sin(x) = opposite / hypotenuse = 11/25 = 0.44

Then:

> x = sin⁻¹(0.44)

Calculate:

sin⁻¹(0.44) ≈ 26.1 degrees → rounds to 26

Options: 26, 64, 24, 77 → 26 is there

✔ Answer for #2 is 26

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Problem 3:

Right triangle with:
- Angle x at the top
- Right angle at bottom right
- Side adjacent to x = 1 (vertical side)
- Side opposite to x = 2 (horizontal side)

Wait — let’s be precise.

Angle x is at the top vertex.

The side next to angle x (adjacent) is the vertical side = 1
The side across from angle x (opposite) is the horizontal side = 2

So:

> tan(x) = opposite / adjacent = 2 / 1 = 2

Then:

> x = tan⁻¹(2)

Calculate:

tan⁻¹(2) ≈ 63.4 degrees → rounds to 63

Options: 63, 99, 35, undefined → 63 is there

✔ Answer for #3 is 63

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Final Answer:
65
26
63