Introduction to Trigonometry · Lesson 2 of 5
Trigonometric Ratios
“Sine, cosine and tangent step in to make right triangles far less mysterious.”
• By the end of this lesson, you should be able to. • Identify opposite, adjacent and hypotenuse with respect to a chosen acute angle. • Define all six trigonometric ratios. • Use reciprocal and quotient relationships between the ratios. • Find remaining ratios when one trigonometric ratio is known. • Explain why a trigonometric ratio depends on the angle, not on the size of the triangle.
Trigonometry begins with a simple question: if we choose one acute angle in a right triangle, how are the three sides related to that angle? The answer is given by six ratios.
First Identify the Three Sides
Consider right triangle ABC, right-angled at B, and focus on angle A. AC is the hypotenuse because it is opposite the 90° angle. BC lies directly across from angle A, so BC is the opposite side. AB touches angle A and is not the hypotenuse, so AB is the adjacent side.
The hypotenuse never changes, but opposite and adjacent do change when you switch from one acute angle to the other. Always mark the angle first, then label the sides.
The Six Trigonometric Ratios
For an acute angle A in a right triangle, the six trigonometric ratios compare pairs of sides. Three are basic ratios — sine, cosine and tangent — and the other three are their reciprocals.
| Ratio | Side relationship |
|---|---|
| sin A | opposite / hypotenuse |
| cos A | adjacent / hypotenuse |
| tan A | opposite / adjacent |
| cosec A | hypotenuse / opposite |
| sec A | hypotenuse / adjacent |
| cot A | adjacent / opposite |
Useful Relationships Between the Ratios
Because the ratios are built from the same three sides, several relationships follow immediately.
sin A means 'sine of angle A'. It is not sin × A. Similarly, sin²A means (sin A)². Also, cosec A = 1/sin A, but sin⁻¹A is not the same thing; inverse trigonometric notation is a different topic studied later.
Why the Ratio Depends Only on the Angle
Imagine several right triangles that all contain the same acute angle A but have different sizes. These triangles are similar by the AA criterion. Corresponding sides of similar triangles are proportional, so opposite/hypotenuse, adjacent/hypotenuse and opposite/adjacent all remain unchanged. Therefore the trigonometric ratios are fixed for a fixed angle.
Problem
In right triangle ABC, right-angled at B, AB = 8 cm, BC = 15 cm and AC = 17 cm. Find all six trigonometric ratios of angle A.
- 1.Relative to angle A: opposite = BC = 15, adjacent = AB = 8, hypotenuse = AC = 17.
- 2.sin A = 15/17.
- 3.cos A = 8/17.
- 4.tan A = 15/8.
- 5.cosec A = 17/15.
- 6.sec A = 17/8.
- 7.cot A = 8/15.
Problem
If tan θ = 5/12, find the other five trigonometric ratios.
- 1.tan θ = opposite/adjacent = 5/12.
- 2.Take opposite = 5k and adjacent = 12k.
- 3.By Pythagoras theorem, hypotenuse = √[(5k)² + (12k)²] = 13k.
- 4.sin θ = 5/13 and cos θ = 12/13.
- 5.cosec θ = 13/5, sec θ = 13/12 and cot θ = 12/5.
For an acute angle, the hypotenuse is the longest side. Therefore sin A and cos A cannot be greater than 1. Their reciprocals, cosec A and sec A, are at least 1 whenever they are defined.
Problem
In right triangle PQR, right-angled at Q, PR = 25 cm and PQ = 7 cm. Find sin P, cos P and tan P.
- 1.PR is the hypotenuse because it is opposite the right angle.
- 2.First find QR using Pythagoras theorem.
- 3.QR² = PR² - PQ² = 25² - 7² = 625 - 49 = 576.
- 4.So QR = 24 cm.
- 5.Relative to angle P: opposite = QR = 24, adjacent = PQ = 7, hypotenuse = PR = 25.
- 6.sin P = 24/25, cos P = 7/25, tan P = 24/7.
Quiz
In triangle ABC, right-angled at B, which side is the hypotenuse?
Which ratio correctly defines cos A in a right triangle?
Which trigonometric ratio is the reciprocal of sin A?
If tan θ = 3/4 and θ is acute, what is sec θ?
Why do trigonometric ratios remain unchanged when a right triangle is enlarged without changing its angles?
Practice Problems
- In △ABC, right-angled at B, AB = 20 cm and BC = 21 cm. Find (i) sin A and cos A, (ii) sin C and cos C.
- If sin A = 5/13, calculate cos A and tan A.
- Given 12 cot A = 5, find sin A and sec A.
- Given sec θ = 17/15, calculate the other five trigonometric ratios.
- In △PQR, right-angled at Q, PR + QR = 37 cm and PQ = 12 cm. Determine sin P, cos P and tan P.
Key Takeaways
• Always identify the reference angle before labeling opposite and adjacent. • sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent. • cosec, sec and cot are reciprocals of sin, cos and tan. • tan A = sin A / cos A and cot A = cos A / sin A. • If one ratio is known, build a right triangle and use Pythagoras theorem to find the others. • For a fixed angle, trigonometric ratios stay the same even if the triangle is scaled.
Next, we will derive and learn the exact trigonometric values of 0°, 30°, 45°, 60° and 90° — and understand where those famous values actually come from.