Trigonometric Ratios
“Trig ratios are a converter that turns an angle into a ratio of sides.”
The formula
sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adjHow to read it: in a right triangle, relative to an angle θ, the ratios of the opposite side, the adjacent side, and the hypotenuse
- sin θ
- — the side opposite θ ÷ the hypotenuse
- cos θ
- — the side adjacent to θ ÷ the hypotenuse
- tan θ
- — the side opposite θ ÷ the side adjacent to θ (= the slope)
The hook
The height of a mountain or the width of a river you can't measure directly — you find it from a single angle and a ratio. That's trigonometry.
In plain words
Pick one acute angle θ in a right triangle, and the ratios between its sides are fixed by the angle alone. Those ratios are sin, cos, and tan.
The intuition
Similar right triangles have the same side 'ratios' even at different sizes. So as long as θ is the same, the triangle can be large or small and sin, cos, tan stay identical. The angle itself sets 'the ratio of the sides'.
How it's built
Name the three sides relative to θ — opposite (rise), adjacent (base), hypotenuse. sin is opposite/hypotenuse, cos is adjacent/hypotenuse, tan is opposite/adjacent. tan is also just the 'slope' of that angle.
Example
In a 3-4-5 right triangle, let θ be the angle opposite the side of length 3: sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, tan θ = 3/4 = 0.75.
Common mistake
It's easy to swap 'opposite' and 'adjacent' in sin and cos. The key is that which side is opposite vs. adjacent depends on where the reference angle θ is.
Where it's used
Measuring mountain heights and distances by triangulation, the grade of a ramp, vibration, waves, sound, even rotation in computer graphics — the bridge between angles and lengths.
Where it came from
Ancient Greek and Indian astronomers built 'tables of chords' to measure distances from the angles of stars above the horizon — the beginning of trigonometry.
Prerequisites
Quick check
What is the value of sin 30°?
- 0.5✓
- 0.707
- 0.866
- 1
Practice
What is the value of tan 45°?
Answer: 1
- 45° comes from a right isosceles triangle — the opposite and adjacent sides are equal
- tan 45° = opposite/adjacent = 1/1 = 1
Takeaway: A tan of 1 means opposite equals adjacent — a 45° slope.
In a 3-4-5 right triangle, let θ be the angle opposite the side of length 3. What is cos θ?
Answer: 0.8
- opposite θ = 3, adjacent to θ = 4, hypotenuse = 5
- cos θ = adjacent/hypotenuse = 4/5
- = 0.8
Takeaway: cos is 'adjacent ÷ hypotenuse' — pick the side touching the reference angle.
Find the value of sin 60° as an approximation. Write on your notepad why it's √3/2.
Answer: 0.87
- Cut an equilateral triangle of side 2 in half to get a 30°-60°-90° triangle
- hypotenuse 2, base 1, height √3
- sin 60° = height/hypotenuse = √3/2 ≈ 0.866
Takeaway: The side ratio of a 30°-60°-90° triangle is 1 : √3 : 2 — worth memorizing.