Divide one side by another
sin, cos and tan are ratios of the sides of a right triangle.
A ratio of two sides
Can we find the height of the Qutub Minar without measuring it? Trigonometry does it by comparing the sides of a right triangle with one of its angles.
To compare two sides we use a ratio, one side divided by another. Take a right triangle ABC with BC = 3 cm, AB = 4 cm and AC = 5 cm.
Which fraction compares the side opposite to angle A (BC) with the hypotenuse (AC)?
What this lesson covers
The idea
For an acute angle A of a right triangle, sin A = side opposite/hypotenuse, cos A = side adjacent/hypotenuse and tan A = side opposite/side adjacent; these are trigonometric ratios of angle A.
A ratio of two sides
Can we find the height of the Qutub Minar without measuring it? Trigonometry does it by comparing the sides of a right triangle with one of its angles.
To compare two sides we use a ratio, one side divided by another. Take a right triangle ABC with BC = 3 cm, AB = 4 cm and AC = 5 cm.
Which fraction compares the side opposite to angle A (BC) with the hypotenuse (AC)?
- 3/5
- 5/3
- 3/4
Read the ratios live
The triangle is drawn to scale (one square is 1 cm). Drag the corner C and watch the three fractions under the picture.
Sine, cosine, tangent
For the acute angle A of a right triangle ABC: sin A = BC/AC cos A = AB/AC tan A = BC/AB
In words: sin A = opposite / hypotenuse, cos A = adjacent / hypotenuse, tan A = opposite / adjacent. These are the sine, cosine and tangent of angle A, called the trigonometric ratios of angle A. For the 3-4-5 triangle: sin A = 3/5, cos A = 4/5 and tan A = 3/4.
The first use of the idea of sine, the way we use it today, is in the *Aryabhatiyam* by Aryabhata, in A.D. 500.
Notes
sin A = BC/AC cos A = AB/AC tan A = BC/AB These are the trigonometric ratios of the acute angle A: sin A = opposite / hypotenuse, cos A = adjacent / hypotenuse, tan A = opposite / adjacent.
Check yourself
A right triangle ABC (right angle at B) has BC = 6 cm, AB = 8 cm and AC = 10 cm. What is sin A?
Answer: 0.6
sin A = BC/AC = 6/10 = 0.6.
Which fraction is cos A in triangle ABC, right-angled at B?
In the same triangle (BC = 6, AB = 8, AC = 10), what is tan A?
Answer: 0.75
tan A = BC/AB = 6/8 = 0.75.
Now look at angle C in the triangle with AB = 12 cm, BC = 5 cm and AC = 13 cm. What is tan C?
Answer: 2.4
For angle C, the opposite side is AB = 12 and the adjacent side is BC = 5. tan C = 12/5 = 2.4.
- BC/AC. That is sin A: the opposite side over the hypotenuse.
- AB/BC. That is the adjacent side over the opposite side. It is not sin, cos or tan of A in this form.
- AB/AC — correct. Yes! cos A = adjacent / hypotenuse = AB/AC.