Trigonometric ratios

Trigonometric Ratios

Trigonometric ratios are essential in understanding the relationships between the angles and sides of triangles, particularly right-angled triangles. In Grade 11, you will focus on the primary trigonometric ratios: sine, cosine, and tangent.

1. Right-Angled Triangles

A right-angled triangle has one angle that measures 90 degrees. The sides of the triangle are referred to as follows:

• Hypotenuse: the longest side, opposite the right angle.
• Opposite side: the side opposite the angle you are considering.
• Adjacent side: the side next to the angle you are considering, which is not the hypotenuse.

2. Trigonometric Ratios

There are three primary trigonometric ratios based on the relationships between the sides of a right-angled triangle:

• Sine (sin): The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse.
• Cosine (cos): The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
• Tangent (tan): The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side.

This can be summarised as follows:

sin(θ) = opposite/hypotenuse
cos(θ) = adjacent/hypotenuse
tan(θ) = opposite/adjacent

3. Example of Trigonometric Ratios

Consider a right-angled triangle where one angle, θ, measures 30 degrees. The lengths of the sides are as follows:

• Opposite side = 3 cm
• Adjacent side = 5 cm
• Hypotenuse = 6 cm

Now, we can calculate the trigonometric ratios for angle θ:

3.1 Sine

Using the sine ratio:

sin(θ) = opposite/hypotenuse

Substituting the values:

sin(30°) = 3/6 = 1/2

3.2 Cosine

Using the cosine ratio:

cos(θ) = adjacent/hypotenuse

Substituting the values:

cos(30°) = 5/6

3.3 Tangent

Using the tangent ratio:

tan(θ) = opposite/adjacent

Substituting the values:

tan(30°) = 3/5

4. Angle of Elevation and Depression

The angle of elevation is the angle formed by the line of sight when looking up from a horizontal line. The angle of depression is the angle formed by the line of sight when looking down from a horizontal line. You can use trigonometric ratios to solve problems involving these angles.

4.1 Example Problem

A person is standing 50 meters away from a building. If the angle of elevation to the top of the building is 60 degrees, find the height of the building.

Let h be the height of the building. Using the tangent ratio:

tan(60°) = opposite/adjacent

Substituting the known values:

tan(60°) = h/50

Now, solve for h:

h = 50 × tan(60°)

Using the value of tan(60°) which is √3:

h = 50 × √3 ≈ 86.6 meters

5. Special Angles

Some angles have known sine, cosine, and tangent values. These angles are 0°, 30°, 45°, 60°, and 90°. Memorising these values can help you solve problems more quickly.

6. Summary

• Trigonometric ratios relate the angles and sides of right-angled triangles.
• The primary ratios are sine, cosine, and tangent.
• Understanding the angles of elevation and depression helps in real-life applications.
• Memorising the trigonometric values for special angles can simplify calculations.

Check your understanding

1. Calculate sin(45°), cos(45°), and tan(45°).
2. A ladder leans against a wall making an angle of 75° with the ground. If the base of the ladder is 2 meters from the wall, find the height at which the ladder touches the wall.
3. What is the sine of an angle if the opposite side is 4 cm and the hypotenuse is 5 cm?
4. In a right triangle, if the adjacent side is 3 cm and the hypotenuse is 6 cm, what is the cosine of the angle?