Full Prep Academy

Overview

Trigonometric ratios are the foundation of trigonometry and are used to relate the angles of a right triangle to the lengths of its sides. The three primary trigonometric ratios are:

• Sine (sin): The ratio of the length of the side opposite the angle 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 side opposite the angle to the length of the adjacent side.

Key Concepts

In a right triangle, the three sides are categorized as follows:

• Opposite side: The side opposite the angle \( \theta \) (not the right angle).
• Adjacent side: The side next to the angle \( \theta \), excluding the hypotenuse.
• Hypotenuse: The longest side of the triangle, opposite the right angle.

Using these definitions, we can express the trigonometric ratios for an angle \( \theta \) in a right triangle as follows:

• sin(θ) = Opposite / Hypotenuse
• cos(θ) = Adjacent / Hypotenuse
• tan(θ) = Opposite / Adjacent

Example

Consider a right triangle with the following side lengths:

• Opposite side = 3 units
• Adjacent side = 4 units
• Hypotenuse = 5 units

The trigonometric ratios for this triangle are:

• sin(θ) = 3 / 5 = 0.6
• cos(θ) = 4 / 5 = 0.8
• tan(θ) = 3 / 4 = 0.75

Practice Questions

1. In a right triangle, the opposite side is 7 units, the adjacent side is 24 units, and the hypotenuse is 25 units. Find:
   • \( \sin(\theta) \)
   • \( \cos(\theta) \)
   • \( \tan(\theta) \)
   Solution

   • \( \sin(\theta) = \frac{7}{25} = 0.28 \)
   • \( \cos(\theta) = \frac{24}{25} = 0.96 \)
   • \( \tan(\theta) = \frac{7}{24} = 0.29 \)
2. A right triangle has a hypotenuse of 13 units and an angle \( \theta = 53^\circ \). The adjacent side is 12 units. Find the length of the opposite side.
   Solution

   Using \( \cos(53^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} \), we can find the opposite side using the sine function:

   First, calculate \( \sin(53^\circ) = 0.798 \), then use \( \text{opposite} = \sin(53^\circ) \times \text{hypotenuse} \)

   \( \text{opposite} = 0.798 \times 13 = 10.37 \) units.

3. Given a right triangle where the opposite side is 8 units and the adjacent side is 15 units, find:
   • \( \sin(\theta) \)
   • \( \cos(\theta) \)
   • \( \tan(\theta) \)
   Solution

   To find the hypotenuse, use the Pythagorean theorem:

   \( \text{hypotenuse} = \sqrt{8^2 + 15^2} = \sqrt{64 + 225} = \sqrt{289} = 17 \)

   • \( \sin(\theta) = \frac{8}{17} = 0.47 \)
   • \( \cos(\theta) = \frac{15}{17} = 0.88 \)
   • \( \tan(\theta) = \frac{8}{15} = 0.53 \)