🧠 Trigonometry Basics

Summary Cheat Sheet

• Six trigonometric ratios connect angles with side lengths in right triangles.
• Standard angle values and identities are high-frequency exam fundamentals.
• Field slope, height, and channel gradient estimation use trigonometric methods.

References

1 source

This lesson introduces trigonometric ratios and identities used in field measurement, slope estimation, and agricultural surveying.

---

Trigonometric Ratios

Trigonometry deals with the relationships between the sides and angles of triangles. In a right-angled triangle with an angle theta, the sides are named relative to that angle: hypotenuse (longest side, opposite the right angle), opposite (side facing the angle), and adjacent (side next to the angle).

The six fundamental trigonometric ratios are:

• sin(theta) = Opposite / Hypotenuse
• cos(theta) = Adjacent / Hypotenuse
• tan(theta) = Opposite / Adjacent = sin(theta) / cos(theta)
• cosec(theta) = 1 / sin(theta) = Hypotenuse / Opposite
• sec(theta) = 1 / cos(theta) = Hypotenuse / Adjacent
• cot(theta) = 1 / tan(theta) = Adjacent / Opposite

Standard Angle Values

Key values to memorize:

Angle	sin	cos	tan
0 degrees	0	1	0
30 degrees	1/2	sqrt(3)/2	1/sqrt(3)
45 degrees	1/sqrt(2)	1/sqrt(2)	1
60 degrees	sqrt(3)/2	1/2	sqrt(3)
90 degrees	1	0	undefined

Important identities:

• sin^2(theta) + cos^2(theta) = 1
• 1 + tan^2(theta) = sec^2(theta)
• 1 + cot^2(theta) = cosec^2(theta)

Heights and Distances

Trigonometry is practically applied to measure heights and distances that cannot be measured directly. Two key terms are:

• Angle of elevation: The angle between the horizontal line of sight and the line of sight directed upward to an object above the observer
• Angle of depression: The angle between the horizontal line of sight and the line of sight directed downward to an object below the observer

Example 1: A farmer standing 30 m from the base of a water tower observes the top at an angle of elevation of 60 degrees. Find the height of the tower.

• tan(60) = height / 30
• sqrt(3) = height / 30
• Height = 30 x sqrt(3) = 30 x 1.732 = 51.96 m

Example 2: From the top of a 20 m high observation post, a field boundary is observed at an angle of depression of 30 degrees. Find the distance from the base of the post to the boundary.

• tan(30) = 20 / distance
• 1/sqrt(3) = 20 / distance
• Distance = 20 x sqrt(3) = 34.64 m

Applications in Surveying

Trigonometry is essential in land surveying and agricultural engineering:

• Slope measurement: Determining the gradient of hillside terraces. If a terrace rises 5 m over a horizontal distance of 50 m, the angle of slope = arctan(5/50) = arctan(0.1) = approximately 5.7 degrees.
• Canal and irrigation design: Calculating the slope (gradient) of irrigation channels for proper water flow. A canal dropping 1 m over 500 m has a gradient of 1:500.
• Land area of sloped fields: The actual cultivable area on a slope is greater than the flat projected area. If a field has a projected area of 1 hectare on a 15-degree slope, the actual surface area = 1 / cos(15) = 1 / 0.966 = 1.035 hectares.
• Tree height estimation: Using clinometers (angle-measuring devices) and trigonometric calculations to estimate tree heights for timber volume assessment in agroforestry.
• Contour mapping: Surveyors use trigonometric levelling to determine elevation differences across agricultural fields for contour bunding and land levelling.

---