Trigonometric ratios are helpful tools that help us solve problems with right triangles. They let us find missing sides and angles easily. In any right triangle, the three most important ratios we use are sine, cosine, and tangent. Let’s take a closer look at each one:

1. Sine (sin): This compares the length of the opposite side to the longest side, called the hypotenuse. It is shown as: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$

2. Cosine (cos): This compares the length of the adjacent side (the side next to the angle) to the hypotenuse: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$

3. Tangent (tan): This compares the opposite side to the adjacent side: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Example:

Let’s look at a right triangle where one angle is 30 degrees and the hypotenuse measures 10 units.

• To find the opposite side, we use the sine function: $\sin(30^\circ) = \frac{\text{opposite}}{10}$ We know that $\sin(30^\circ) = 0.5$, so we can rearrange the equation to find the opposite side: $\text{opposite} = 0.5 \times 10 = 5 \text{ units}$

• Now, to find the adjacent side, we use the cosine function: $\cos(30^\circ) = \frac{\text{adjacent}}{10}$ We know that $\cos(30^\circ) = \frac{\sqrt{3}}{2}$, so we can find: $\text{adjacent} = \frac{\sqrt{3}}{2} \times 10 \approx 8.66 \text{ units}$

By using these ratios, you can easily find any missing sides or angles in a right triangle!