1.1 Trigonometric Ratios

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles, and trigonometric ratios are the fixed ratios between the lengths of two sides of a right-angled triangle for a given angle. These ratios allow unknown lengths and angles to be calculated from a few measurements, which is why they are used daily in construction, surveying, roofing, drainage and mechanical work. Once the ratios are understood, technicians can work out roof pitches, rafter lengths, ramp slopes, heights of buildings and distances that cannot be measured directly. Example: a carpenter preparing roof trusses for a classroom block at a primary school in Kitui used trigonometric ratios to calculate the rafter length and rise from the span and roof pitch shown on the drawing.

1.1.1 Sine, Cosine, Tangent, Cosecant, Secant, Cotangent

In a right-angled triangle, the side opposite the right angle is the hypotenuse (the longest side), the side facing the angle being considered (θ) is the opposite side, and the side next to the angle (other than the hypotenuse) is the adjacent side. The three basic ratios, sine (sin), cosine (cos) and tangent (tan), compare these sides, and their reciprocals, cosecant (cosec), secant (sec) and cotangent (cot), are obtained by turning each basic ratio upside down. The value of each ratio depends only on the angle, not on the size of the triangle, so a 30° roof gives the same ratios whether the building is small or large. Example: when the foreman at a hotel construction site in Malindi checked a staircase, he used the tangent of the stair angle to relate the riser height to the tread length.

The Six Trigonometric Ratios

A useful memory aid for the basic ratios is SOH CAH TOA: Sine = Opposite ÷ Hypotenuse, Cosine = Adjacent ÷ Hypotenuse, Tangent = Opposite ÷ Adjacent.

Important Relationships Between the Ratios

• Quotient relationship: tan θ = sin θ ÷ cos θ, and cot θ = cos θ ÷ sin θ.
• Pythagorean identity: sin² θ + cos² θ = 1, which comes directly from Pythagoras’ theorem.
• Related identities: 1 + tan² θ = sec² θ, and 1 + cot² θ = cosec² θ.
• Complementary angles: sin θ = cos (90° − θ), so the sine of one acute angle equals the cosine of the other.
• Range of values: sin θ and cos θ always lie between 0 and 1 for acute angles, while tan θ can be any positive value.

Ratios of Special Angles

Steps to Find Trigonometric Ratios in a Right-Angled Triangle

1. Identify the right angle and the hypotenuse: The hypotenuse is opposite the right angle and is the longest side.
2. Mark the angle θ: Identify the angle whose ratios are required.
3. Label the opposite and adjacent sides: Name the sides relative to θ.
4. Find any missing side: Use Pythagoras’ theorem, hypotenuse² = opposite² + adjacent².
5. Write the basic ratios: Calculate sin θ, cos θ and tan θ from the side lengths.
6. Write the reciprocal ratios: Invert the basic ratios to get cosec θ, sec θ and cot θ.
7. Check the results: Confirm that sin² θ + cos² θ = 1.

Worked Example 1: All Six Ratios

A right-angled triangle has an opposite side of 3 m, an adjacent side of 4 m and a hypotenuse of 5 m. Find all six ratios of θ.

sin θ = Opposite ÷ Hypotenuse

sin θ = 3 ÷ 5

sin θ = 0.6

cos θ = Adjacent ÷ Hypotenuse

cos θ = 4 ÷ 5

cos θ = 0.8

tan θ = Opposite ÷ Adjacent

tan θ = 3 ÷ 4

tan θ = 0.75

cosec θ = 1 ÷ sin θ = 5 ÷ 3 = 1.667

sec θ = 1 ÷ cos θ = 5 ÷ 4 = 1.25

cot θ = 1 ÷ tan θ = 4 ÷ 3 = 1.333

Check: sin² θ + cos² θ = 0.36 + 0.64 = 1

Worked Example 2: Finding the Other Ratios from One Ratio

Given sin θ = 5 ÷ 13 for an acute angle, find cos θ, tan θ and θ.

Let the opposite side = 5 and the hypotenuse = 13.

Adjacent = √(Hypotenuse² − Opposite²)

Adjacent = √(13² − 5²) = √(169 − 25) = √144

Adjacent = 12

cos θ = 12 ÷ 13 = 0.923

tan θ = 5 ÷ 12 = 0.417

θ = sin⁻¹(5 ÷ 13)

θ = sin⁻¹(0.3846)

θ = 22.6°

Using a Scientific Calculator

• Set the angle mode: Make sure the calculator is in degree mode (D or DEG) for angles in degrees.
• Find a ratio: Press the ratio key and enter the angle, for example sin 35 gives 0.5736.
• Find an angle from a ratio: Use the inverse keys sin⁻¹, cos⁻¹ or tan⁻¹ (usually SHIFT then the ratio key).
• Reciprocal ratios: Calculate cosec, sec and cot as 1 ÷ sin, 1 ÷ cos and 1 ÷ tan, since most calculators have no separate keys.
• Round sensibly: Keep at least four decimal places in working and round only the final answer.

1.1.2 Applications of right-angle trigonometry

Right-angle trigonometry is used whenever a practical problem can be represented by a right-angled triangle, such as a roof slope, a ramp, a staircase, a sloping drain, a ladder against a wall or a line of sight to the top of a building. By measuring one side and one angle, or two sides, the remaining sides and angles can be calculated without measuring them directly, which saves time and makes it possible to find heights and distances that are hard to reach. Common terms include the angle of elevation, measured upwards from the horizontal to an object above the observer, and the angle of depression, measured downwards from the horizontal to an object below. Example: the site engineer for a hospital extension in Nyeri used the angle of elevation measured with a clinometer to estimate the height of an existing building before planning crane lifts beside it.

Instruments for Measuring Angles and Distances

Common Applications in Construction

• Roof construction: Calculating rafter lengths, roof rise and pitch from the span and angle.
• Ramps and access: Designing ramps for wheelchair access to the required gradient and length.
• Staircases: Relating riser height, tread length and stair angle for safe and comfortable stairs.
• Heights of structures: Finding heights of buildings, trees and towers using angles of elevation.
• Drainage and pipe gradients: Calculating the fall of drains and pipes over a given length.
• Bracing and struts: Calculating lengths and angles of diagonal braces in frames and scaffolds.
• Setting out: Laying out angles and offsets on site during surveying and setting out.

Steps for Solving Practical Right-Angle Problems

1. Draw a sketch: Represent the problem with a right-angled triangle showing all known values.
2. Identify the right angle: Make sure the triangle used actually contains a right angle.
3. Label the sides: Mark the hypotenuse, opposite and adjacent sides relative to the known or required angle.
4. Choose the correct ratio: Select the ratio that links the known values to the unknown, using SOH CAH TOA.
5. Write the equation: Substitute the known values into the ratio.
6. Solve for the unknown: Rearrange and calculate the missing side or angle.
7. Add other heights or offsets: Include instrument heights or other dimensions where needed.
8. Check the answer: Confirm the answer is reasonable and has the correct units.

Worked Example 3: Rafter Length and Roof Rise

A roof has a span of 8 m and a pitch of 30°. Find the length of each rafter (ignoring overhang) and the rise of the roof.

Half span (adjacent) = 8 ÷ 2 = 4 m

Rafter length = Adjacent ÷ cos θ

Rafter length = 4 ÷ cos 30°

Rafter length = 4 ÷ 0.866

Rafter length = 4.62 m

Rise = Adjacent × tan θ

Rise = 4 × tan 30°

Rise = 4 × 0.577

Rise = 2.31 m

Worked Example 4: Cost of Rafters

The roof in Example 3 is 12 m long, with trusses at 1.2 m centres, and each truss has two rafters. Rafter timber costs Ksh 250 per metre. Find the cost of the rafter timber.

Number of trusses = (Length ÷ Spacing) + 1

Number of trusses = (12 ÷ 1.2) + 1

Number of trusses = 11

Total rafter length = Trusses × Rafters per truss × Rafter length

Total rafter length = 11 × 2 × 4.62

Total rafter length = 101.64 m

Cost = Total length × Rate

Cost = 101.64 × 250

Cost = Ksh 25,410

Worked Example 5: Height of a Building

A surveyor stands 30 m from the base of a building and measures an angle of elevation of 35° to the top. The instrument is 1.5 m above the ground. Find the height of the building.

Height above instrument = Distance × tan θ

Height above instrument = 30 × tan 35°

Height above instrument = 30 × 0.700

Height above instrument = 21.0 m

Building height = 21.0 + 1.5

Building height = 22.5 m

Worked Example 6: Access Ramp

An access ramp at the entrance of a SACCO office in Meru must rise 0.6 m at a gradient of 1 in 12. Find the angle of the ramp and its sloping length.

θ = tan⁻¹(Rise ÷ Run)

θ = tan⁻¹(1 ÷ 12)

θ = 4.76°

Sloping length = Rise ÷ sin θ

Sloping length = 0.6 ÷ sin 4.76°

Sloping length = 0.6 ÷ 0.0830

Sloping length = 7.23 m

Worked Example 7: Angle of Depression

From the top of a water tower 15 m high at a county water office in Machakos, the angle of depression to a vehicle on level ground is 25°. Find the horizontal distance of the vehicle from the foot of the tower.

Distance = Height ÷ tan θ

Distance = 15 ÷ tan 25°

Distance = 15 ÷ 0.4663

Distance = 32.17 m

Chapter Summary

This chapter covered the fundamental trigonometric ratios including sine, cosine, tangent, cosecant, secant, and cotangent, with emphasis on their use in right-angle triangle problems. It explored key trigonometric identities such as the Pythagorean identity and angle sum and difference formulas, as well as double and half-angle identities, which are essential tools for simplifying expressions. The chapter then demonstrated how to apply these identities to solve trigonometric equations, highlighting their practical applications in real-world scenarios. Moving beyond circular functions, the chapter introduced hyperbolic functions including sinh, cosh, and tanh, along with their defining identities. It further examined methods for solving equations involving hyperbolic functions, emphasizing their relevance in engineering and physics contexts. Together, these topics provide a comprehensive understanding of both trigonometric and hyperbolic functions, equipping learners to tackle complex problems in mathematics and applied sciences.