To find the unknown sides of a right triangle using trigonometry, just follow these simple steps:

Step 1: Identify the Triangle

Start by figuring out which triangle you’re working with. A right triangle has one angle that is exactly $90^\circ$.

Step 2: Label the Sides

Next, label the sides of the triangle based on the angle you’re focusing on:

• Hypotenuse ($c$): This is the side across from the right angle. It's the longest side.
• Adjacent Side ($a$): This side is next to the angle you're working with.
• Opposite Side ($b$): This side is across from the angle you are studying.

Step 3: Choose the Right Trigonometric Function

Now, depending on what information you already have (angles and sides), pick the right trigonometric function:

• Sine: $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{b}{c}$
• Cosine: $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{a}{c}$
• Tangent: $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{b}{a}$

Step 4: Rearrange the Equation

If you have the angle ($\theta$) and one side, you can change the equation to find the unknown side:

• If you want to find the opposite side ($b$): $b = c \cdot \sin(\theta)$
  or $b = a \cdot \tan(\theta)$
• If you're looking for the adjacent side ($a$): $a = c \cdot \cos(\theta)$
  or $a = \frac{b}{\tan(\theta)}$
• If you need the hypotenuse ($c$): $c = \frac{b}{\sin(\theta)}$
  or $c = \frac{a}{\cos(\theta)}$

Step 5: Plug in Known Values

Now, plug in the values you know into the new equation. For example, if $\theta = 30^\circ$ and the opposite side $b = 5$, you can find the hypotenuse $c$ like this: $c = \frac{5}{\sin(30^\circ)}$
Since $\sin(30^\circ) = 0.5$, that means:
$c = \frac{5}{0.5} = 10.$

Step 6: Calculate the Unknown Side

Carefully do the math using a calculator. Just make sure your angles are in the right mode (degrees or radians).

Conclusion

By following these easy steps, you can use trigonometric functions to find the unknown sides of right triangles. This can be really helpful in lots of areas like construction and navigation!