A trigonometric ratio is a ratio of two side lengths in a right triangle, and once the size of one acute angle is fixed, the values of sin, cos and tan are fixed too, whatever the size of the triangle.
Right triangles of different sizes with the same acute angle are similar, so their side ratios are the same.
This is why the height of a building or a tree can be calculated from the distance to it and the angle of elevation, without climbing it.
The picture draws a right triangle with hypotenuse 1, so as the angle θ changes, the height changes as sin θ and the base as cos θ.
Side ratios of a right triangle
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The height sin θ, the base cos θ and the value of tan θ in a right triangle with hypotenuse 1 as θ changes
30°
📐 Definitions
①sin θ = height ÷ hypotenuse: the side opposite θ divided by the hypotenuse.
②cos θ = base ÷ hypotenuse: the side next to θ divided by the hypotenuse.
③tan θ = height ÷ base.
④If the hypotenuse is 1, the height is sin θ itself and the base is cos θ itself, which is exactly the case in the picture.
Ratios of the special angles (30°, 45°, 60°)
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A table of sin, cos and tan for 30°, 45° and 60° side by side
45°
⭐ Values for the special angles
①At 30°, sin = 1/2, cos = √3/2 and tan = √3/3.
②At 45°, sin = cos = √2/2 and tan = 1.
③At 60°, sin = √3/2, cos = 1/2 and tan = √3.
④The sin and cos values of 30° and 60° swap, because the two angles add up to 90°, so the height for one angle is the base for the other.
Relations between the ratios
tan in terms of sin and cos
tan θ = sin θcos θ
tan θ is sin θ divided by cos θ.
Relation from Pythagoras' theorem
sin²θ + cos²θ = 1
This is Pythagoras' theorem applied to a right triangle with hypotenuse 1.
💡 Using the relations
①If sin θ is known, cos θ = √(1 − sin²θ).
②Knowing one ratio is enough to find the other two.
③For 0° < θ < 90°, sin, cos and tan are all positive.
④The value of sin² + cos² on the right of the picture stays 1 however the angle changes.
Finding heights and distances
Finding a height
height h = d × tan θ
Calculate the height from the distance d and the angle of elevation θ.
🏢 Everyday examples
①Standing 50 m from a building and looking up to its top at 30°, h = 50 × tan 30° ≈ 28.9 m.
②The heights of trees, towers and mountains are found the same way.
③If the measurement is taken at eye level, add the eye height to the result.
④Angles of depression are handled the same way, by forming a right triangle.
Worked examples
Example 1
Find the value of sin 30°.
1
30° is a special angle, so recall its value.
30° → sin=1/2
2
In this right triangle the height is half the hypotenuse, so sin 30° = 1/2.
sin30° = 1/2
▸ 1/2
The values for 30°, 45° and 60° can be read straight from an equilateral triangle and a right isosceles triangle.
Example 2
In a right triangle, sin A = 3/5 and the hypotenuse is 10. Find the side opposite angle A.
1
sin A = opposite ÷ hypotenuse, so opposite = hypotenuse × sin A.
sin A = opp/hyp
2
10 × 3/5 = 6.
opp = 10 × 3/5 = 6
▸ 6
Using the definition of sin backwards gives the opposite side from the hypotenuse and the angle.
Exam checklist
Definitions of the ratios
sin = opphyp, cos = adjhyp, tan = oppadj
Read the height and the base relative to the angle θ; when the angle changes, so do they.
G9 school-exam type
What is the value of tan 60°?
①√3/3
②1/2
③√2
④√3
⑤2
▸ ④ √3
1
Read the special-angle table for 60°.
60° → tan=√3
2
So tan60° = √3.
tan60° = √3
🎯 Exam points
①Know the values for 30°, 45° and 60° exactly.
②Use sin²θ + cos²θ = 1 to find the other ratios.
③Use tan θ = sin θ ÷ cos θ.
④In height and distance problems, choose the ratio from the side you know and the side you want.
⑤At this level only angles from 0° to 90° are used.