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Sin 60 degrees is one of the three important ratios that are used to calculate the length or the angles that are formed by the right-angled triangles. The three important ratios include sin, cos, and tan that is sine, cosine, and tangent respectively. The angle (formed by the hypotenuse and adjacent side) and the opposing side to the angle and hypotenuse are defined by the sine function. Or, it can be said that, the Sine of Angle Theta is equal to the ratio of a right-angled triangle perpendicular and hypotenuse.
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Keywords: Sin 60°, right-angled triangle, perpendicular, hypotenuse, angle, radian
Sine or Sin
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According to the property of a right-angled triangle, if an angle is 90° then the sum of the remaining two angles is the same as the third angle. The major angles that can be noted are 0°, 30°, 45°, 60°, and 90°. Sine, however, implies the ratio between the perpendicular of the right-angled triangle to that of the hypotenuse of the right-angled triangle. One can find the value of 60° either with the help of trigonometric functions or unit circles.
Also read: Introduction to Trigonometry Important Questions
Value of Sin 60°
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Sin defines the ratio between the perpendicular of the right-angled triangle to that of the hypotenuse of the right-angled triangle. In decimal form, its value is 0.8660254.
Sin θ = opposite side /hypotenuse = perpendicular/hypotenuse
| Radian | 0° | π/6 or 30° | π/4 or 45° | π/3 or 60° | π/2 or 90° | π or 90° | 3π/2 or 270° | 2π or 360° |
|---|---|---|---|---|---|---|---|---|
| Sin | 0 | 1/2 | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| Cos | 1 | √3/2 | 1/√2 | 1/2 | 0 | -1 | 0 | 1 |
| Tan | 0 | 1/√3 | 1 | √3 | Not Defined | 0 | Not Defined | 0 |
The values of trigonometry ratios listed above are in degrees. We can also write the values in terms of radians. Radians are used to describe a unit circle with a radius of one
Also read: Some Applications of Trigonometry: Heights and Distances
Sin 60° using Unit Circle
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Follow the steps given below to find the value of Sin 60° using the Unit Circle.
- Using the positive x-axis, rotate the ‘r’ in an anticlockwise direction so that a 60° angle is formed.
- The co-ordinate y which is of the Sin 60° is then equal to the value of 0.866 and the intersection point of the unit circle and r. (0.5,0.866)
- This gives us the Sin 60°= 0.866.
Degrees and Radian
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The two ways to measure any angle include degree and radian. Radians are depicted using while to denote degrees, we use ‘°’. A circle is of 360° that is 2 radians as one is the same as 180°. The degree is then categorized into minutes and seconds.
Trigonometric Functions related to Sin 60°
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Sin 60° will always yield a positive value as it lies in the first quadrant.
Some of the formulas represented by Sin 60° are:
- ± √(sec²(60°) - 1)/sec 60°
- ± √(1-cos²(60°))
- ± 1/√(1 + cot²(60°))
- 1/cosec 60°
- ± tan 60°/√(1 + tan²(60°))
- sin(180° - 60°) = sin 120°
- cos(90° - 60°) = cos 30°
- -sin(180° + 60°) = -sin 240°
- -cos(90° + 60°) = -cos 150°
Things to Remember
- Sin function can be defined as the ratio in a right-angled triangle between its two sides that is perpendicular to that of the hypotenuse.
- In fractional form, the value of sin 60°= √3/2
- Sin 60°, when denoted in the terms of a radian, is π/3.
- The two ways by which the value of the sin 60° can be predicted are by either using the trigonometric functions or by using the unit circle.
- A radian is equal to 180° which is denoted a semi-circle while 2π depicts a full circle.
Sample Questions
Ques. Calculate 2 × (Sin 30° Cos 30°). [Sin 60°= 0.866]. [3 marks]
Ans. To find = 2 × (Sin 30° Cos 30°)
As we know, Sin 2α = 2 SinαCosα
Sin 2 × 30° = 2 Sin 30°Cos 30°
Sin 60° = 2 Sin 30°Cos 30° ---------(1)
Given: Sin 60° = 0.866 ---------(2)
Therefore, from (1) and (2)
2 Sin 30° Cos 30° = 0.866
Ques. Simplify the following: 2 (sin 60°/sin 420°) [3 marks]
Ans. To find: 2 (sin 60°/sin 420°)
Although,
sin 60° = sin 420°
So, 2 sin 60° / sin 420° = 2(sin 60° / sin 60°)
= 2 × 1 = 2
Ques. Calculate Sin 60° when the value of Cosec 60° is 1.1547. [3 marks]
Ans. To find Sin 60°
Given: Cosec 60°= 1.1547
According to the trigonometric formula:
Sin 60° = 1/Cosec 60°
Sin 60° = 1/1.1547
Sin 60° = 0.866
Ques. Calculate the value of α if 2 Sin (60° – α) = 1. [3 marks]
Ans. Given: 2 Sin (60° – α) = 1
Sin (60° – α) = 1/2
Sin (60° – α) = Sin 45°
60° – α = 45°
60° – 45° = α
α = 60° – 45°
α = 15°
Ques. State the formulas of the Sin function that gives the value of Sin 60° in terms of different quadrants. [3 marks]
Ans. The formulas that yield the value of Sin 60° in terms of quadrants are:
- sin(180° – 60°) = sin 120°
- cos(90° – 60°) = cos 30°
- -sin(180° + 60°) = -sin 240°
- -cos(90° + 60°) = -cos 150°
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