There are three sides of a triangle: Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratios. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).
As given in the figure in a right-angle triangle:
1. The side opposite the right angle is called the hypotenuse
2. The side opposite to an angle is called the opposite side
- For angle C opposite side is AB
- For angle A opposite side is BC
3. The side adjacent to an angle is called the adjacent side
- For angle C adjacent side is BC
- For angle A adjacent side is AB
Mathematical symbol θ is used to denote the angle.
Sine (sin)
Sine of an angle is defined by the ratio of lengths of sides which is opposite to the angle and the hypotenuse. It is represented as sin θ.
Cosine (cos)
Cosine of an angle is defined by the ratio of lengths of sides which is adjacent to the angle and the hypotenuse. It is represented as cos θ
Tangent (tan)
Tangent of an angle is defined by the ratio of the length of sides which is opposite to the angle and the side which is adjacent to the angle. It is represented as tan θ
Cosecant (cosec)
Cosecant of an angle is defined by the ratio of the length of the hypotenuse and the side opposite the angle. It is represented as cosec θ
Secant (sec)
Secant (sec θ) is a trigonometric function defined as the ratio of the length of the hypotenuse to the length of the adjacent side in a right-angled triangle. It is the reciprocal of cosine.
Cotangent (cot)
Cotangent of an angle is defined by the ratio of the length of sides that is adjacent to the angle and the side which is opposite to the angle. It is represented as cot θ.
Values of Trigonometric Ratios for Some Specific Angles
Values of trigonometric ratios for some specific angles, which are commonly used in mathematics, are given in the following trigonometric table:
| Angle (in Degree) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
|---|---|---|---|---|---|---|---|---|
| Angle (in Radians) | 0 | π/6 | π/4 | π/3 | π/2 | π | 3π/2 | 2π |
| sin(θ) | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| cos(θ) | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| tan(θ) | 0 | 1/√3 | 1 | √3 | undefined | 0 | undefined | 0 |
| csc(θ) | undefined | 2 | √2 | 2/√3 | 1 | undefined | -1 | undefined |
| sec(θ) | 1 | 2/√3 | √2 | 2 | undefined | -1 | undefined | 1 |
| cot(θ) | undefined | √3 | 1 | 1/√3 | 0 | undefined | 0 | undefined |
Related Articles:
Solved Examples
Example 1: Find the value of sec2 θ if tan2 θ = 1
Given, tan2 θ = 1 ...(1)
we know that, sec2 θ - tan2 θ = 1...(2)
By eq (1),
sec2 θ - 1 = 1
⇒ sec2 θ = 1 + 1Than, sec2 θ = 2
Example 2: Find the value of cos θ if tan θ = √3 and sin θ = √3/2.
Given, tan θ = √3 and sin θ = √3/2
We know that, tan θ = sin θ/cos θ
⇒ √3 = (√3/2)/cos θ
⇒ cos θ = (√3/2)/√3
⇒ cos θ = 1/2
Example 3: Find the value of sin θ if tan θ = 4/3 and cos θ = 6/10.
Given, tan θ = 4/3 and cos θ = 6/10
We know that, tan θ = sin θ/cos θ
⇒ 4/3 = sin θ/(6/10)
⇒ sin θ = (4/3) × (6/10)
⇒ sin θ = 8/10
Example 4: In a right-angled triangle PQR, right-angled at Q, the hypotenuse PR = 13 cm, base QR = 5 cm, and perpendicular PQ = 12 cm.
If ∠PRQ = θ, find sinθ, cosθ, and tanθ.
Given, in ∆PQR:
PR = 13 cm (Hypotenuse)
QR = 5 cm (Base)
PQ = 12 cm (Perpendicular)For angle θ at R:
sinθ = Perpendicular / Hypotenuse = PQ / PR = 12 / 13
cosθ = Base / Hypotenuse = QR / PR = 5 / 13
tanθ = Perpendicular / Base = PQ / QR = 12 / 5