If You Are Given the Hypotenuse and an Adjacent Side: Find the Missing Side
Table of Contents :
To solve problems involving right triangles, knowing the lengths of certain sides can help us find the unknown side using basic trigonometry. If you have the hypotenuse and one adjacent side, you can easily find the missing side (the opposite side) using some well-known relationships. Let's dive deeper into this concept!
Understanding Right Triangles 🔺
A right triangle has one angle that is exactly 90 degrees. The sides are typically referred to as:
- Hypotenuse: The longest side, opposite the right angle.
- Adjacent side: The side next to the angle you are interested in (but not the hypotenuse).
- Opposite side: The side opposite the angle you are working with.
In this case, we will focus on finding the length of the opposite side when the hypotenuse (c) and an adjacent side (a) are known.
The Relationship Between the Sides
To find the opposite side (b), we can use the following trigonometric functions: sine and cosine.
The Pythagorean Theorem 🧮
Before diving into sine and cosine, it's worth mentioning the Pythagorean theorem:
[ c^2 = a^2 + b^2 ]
Where:
- ( c ) is the length of the hypotenuse.
- ( a ) is the length of the adjacent side.
- ( b ) is the length of the opposite side.
However, since we are focusing on finding the opposite side, we can rearrange the formula for our needs.
Using Trigonometric Functions
Step 1: Use Cosine to find the angle ( \theta )
The cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the length of the hypotenuse:
[ \cos(\theta) = \frac{a}{c} ]
From this, we can find the angle ( \theta ):
[ \theta = \cos^{-1} \left( \frac{a}{c} \right) ]
Step 2: Use Sine to find the opposite side
Now, using the sine function, we can find the length of the opposite side:
[ \sin(\theta) = \frac{b}{c} ]
Rearranging this gives us:
[ b = c \cdot \sin(\theta) ]
Example Table
Here’s a quick reference table to summarize the calculations:
| Variable | Description |
|---|---|
| ( c ) | Length of the hypotenuse |
| ( a ) | Length of the adjacent side |
| ( b ) | Length of the opposite side |
| ( \theta ) | Angle between adjacent and hypotenuse |
| Cosine | ( \cos(\theta) = \frac{a}{c} ) |
| Sine | ( \sin(\theta) = \frac{b}{c} ) |
Important Note: “Ensure that your calculator is set to the correct mode (degrees or radians) based on the angle measurement you are using.”
Example Calculation
Let's say we have the following values:
- Hypotenuse (( c )): 10 units
- Adjacent side (( a )): 8 units
Step 1: Find the angle ( \theta )
[ \cos(\theta) = \frac{8}{10} = 0.8 ]
Calculating ( \theta ):
[ \theta = \cos^{-1}(0.8) \approx 36.87^\circ ]
Step 2: Find the opposite side ( b )
Now we can use the sine function:
[ b = 10 \cdot \sin(36.87^\circ) \approx 6 ]
Conclusion
Finding the missing side of a right triangle when given the hypotenuse and one adjacent side is a straightforward application of trigonometric principles. With the Pythagorean theorem and sine and cosine functions, you can determine the length of the opposite side effectively. 🌟
By mastering these techniques, you can tackle a variety of problems involving right triangles with confidence!
