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Calculate the opposite and adjacent sides of the triangle shown below. There is change in the hypotenuse of 85mm 

In this triangle, we are dealing with a right-angled triangle where:

- The hypotenuse is 85 mm.
- The angle given is \( 30^\circ \).

Using trigonometry, we can calculate both the **opposite** and **adjacent** sides of this right triangle.

### Step 1: Finding the Opposite Side
To find the opposite side, we use the sine function:
\[
\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}
\]
Substituting the values:
\[
\sin(30^\circ) = \frac{\text{Opposite}}{85}
\]
We know that \( \sin(30^\circ) = 0.5 \), so:
\[
0.5 = \frac{\text{Opposite}}{85}
\]
Solving for the opposite:
\[
\text{Opposite} = 0.5 \times 85 = 42.5 \text{ mm}
\]

### Step 2: Finding the Adjacent Side
For the adjacent side, we use the cosine function:
\[
\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}
\]
Substituting the values:
\[
\cos(30^\circ) = \frac{\text{Adjacent}}{85}
\]
We know that \( \cos(30^\circ) \approx 0.866 \), so:
\[
0.866 = \frac{\text{Adjacent}}{85}
\]
Solving for the adjacent:
\[
\text{Adjacent} = 0.866 \times 85 = 73.61 \text{ mm}
\]

### Final Answer:
- Opposite side = 42.5 mm
- Adjacent side = 73.61 mm

Let me know if you want further details or if you have any questions!

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Here are five related questions to expand your understanding:
1. How do we apply the tangent function to find the ratio of opposite and adjacent sides?
2. What would happen to the opposite side if the hypotenuse were doubled?
3. How do you calculate the area of this triangle using the sides found?
4. How do the sine and cosine functions relate to the unit circle?
5. Can you calculate the length of the hypotenuse if only the adjacent side and angle were given?

**Tip:** For triangles, knowing at least one angle and one side allows you to solve the entire triangle using basic trigonometric functions!

Calculate the opposite and adjacent sides of the triangle shown below. There is a change in the hypotenuse to 85 mm.

Learn how to calculate the opposite and adjacent sides of a right-angled triangle with a 30° angle and a hypotenuse of 85 mm. Using trigonometric functions, we derive the opposite side as 42.5 mm and the adjacent side as 73.61 mm. Ideal for students in Grades 9-12.

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