Basic trigonometric ratios

Basic Trigonometric Ratios

Trigonometric ratios are the relationships between the angles and sides of a right triangle. These ratios are fundamental in trigonometry and are used to solve various problems involving angles and distances.

Understanding Right Triangles

A right triangle is a triangle that has one angle measuring 90 degrees. The sides of a right triangle are referred to as:

• Hypotenuse: The longest side of the triangle, opposite the right angle.
• Opposite Side: The side opposite the angle of interest.
• Adjacent Side: The side next to the angle of interest, excluding the hypotenuse.

Consider a right triangle where one angle is θ (theta). The sides can be labelled as follows:

          |\
          | \
   Opposite |  \ Hypotenuse
          |   \
          |____\
           Adjacent

Basic Trigonometric Ratios

There are three primary trigonometric ratios:

1. Sine (sin): The ratio of the length of the opposite side to the length of the hypotenuse.
2. Cosine (cos): The ratio of the length of the adjacent side to the length of the hypotenuse.
3. Tangent (tan): The ratio of the length of the opposite side to the length of the adjacent side.

Formulas for Trigonometric Ratios

The formulas for these ratios can be expressed as:

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

Example of Calculating Trigonometric Ratios

Consider a right triangle where the length of the opposite side is 3 units, the length of the adjacent side is 4 units, and the hypotenuse can be calculated using the Pythagorean theorem:

Using the Pythagorean theorem:

c² = a² + b²

where c is the hypotenuse, a is the opposite side, and b is the adjacent side.

c² = 3² + 4²
c² = 9 + 16
c² = 25
c = √25 = 5

Now we can calculate the trigonometric ratios:

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

Special Angles in Trigonometry

There are specific angles in trigonometry that have known sine, cosine, and tangent values. These angles are 0°, 30°, 45°, 60°, and 90°. Here are their corresponding values:

Applications of Trigonometric Ratios

Trigonometric ratios are used in various fields, such as physics, engineering, and architecture. For example, they can be used to determine the height of a building when the distance from the observer is known, or to calculate angles in construction.

Example Application

Suppose you want to find the height of a tree. You stand 10 meters away from the base of the tree and measure the angle of elevation to the top of the tree as 30°. You can use the tangent ratio to find the height:

tan(30°) = Height / Distance

Let Height be h. Then:

tan(30°) = h / 10

Using the known value of tan(30°) = √3/3:

√3/3 = h / 10

Now, solve for h:

h = 10 × (√3/3)
h ≈ 5.77 meters

Common Mistakes

Summary

• Trigonometric ratios include sine, cosine, and tangent.
• These ratios are defined in terms of the sides of a right triangle.
• Special angles have known sine, cosine, and tangent values.
• Trigonometric ratios have various applications in real-life scenarios.

Check your understanding

1. What is the sine of a 45° angle?
2. How do you calculate the height of an object using trigonometric ratios?
3. What are the values of sine, cosine, and tangent for a 60° angle?
4. Identify the opposite and adjacent sides in a right triangle when given a specific angle.