Trigonometric Ratios in Right Triangles

In a right triangle, the trigonometric ratios relate the angles to the lengths of the sides. The primary ratios are:

• Sine ($\sin$): the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine ($\cos$): the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent ($\tan$): the ratio of the length of the opposite side to the length of the adjacent side.

These ratios can be remembered using the acronym SOH-CAH-TOA:

• SOH: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$
• CAH: $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$
• TOA: $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}$

Example Problem

Find the sine, cosine, and tangent of a right triangle with an angle $\theta$ where the opposite side is 3 units and the hypotenuse is 5 units.

1. Identify the sides:
   Opposite = 3, Hypotenuse = 5
2. Calculate sine:
   $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}$
3. Calculate cosine:
   First, find the adjacent side using the Pythagorean theorem:
   $\text{adjacent}^{2} + 3^{2} = 5^{2}$
   $\text{adjacent}^{2} + 9 = 25$
   $\text{adjacent}^{2} = 16$
   $\text{adjacent} = 4$
   Now calculate cosine:
   $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}$
4. Calculate tangent:
   $\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}$

Thus, for the angle $\theta$:

• $\sin(\theta) = \frac{3}{5}$
• $\cos(\theta) = \frac{4}{5}$
• $\tan(\theta) = \frac{3}{4}$

Key Questions

• What is the sine of an angle in a right triangle?
• How do you find the cosine ratio?
• What relationship does the tangent ratio have with the opposite and adjacent sides?
• How can the Pythagorean theorem be used to find the missing side?
• What are some real-world applications of trigonometric ratios?

Keywords

• Sine
• Cosine
• Tangent

• Right Triangle
• Trigonometric Ratios