Understanding Sin 60 Degrees – Definition, Value, Formula & FAQs

In trigonometry, the main ratios are sine (sin), cosine (cos), and tangent (tan), and they are mostly used to measure angles. Some common angles you’ll often see are 0°, 30°, 45°, 60°, 90°, and 180°. Among these, the sine of 60 degrees is one of the important values. You can remember these values more easily by using a trigonometry table. In this explanation, we’ll focus on learning the value of sin 60° and how to use it.

Further Reading:

• Understanding Sine Function
• Sin 0 Degree Explained
• A Look at Sin 30 Degrees
• Delving into Sin 45 Degrees
• Understanding Sin 90 Degrees
• The Law of Sines

Calculating the Value of Sin 60 Degree

In a right-angled triangle, the sine of an angle α is the ratio of the length of the side opposite the angle (perpendicular) to the length of the longest side (hypotenuse).

Sin α= Opposite Side/Hypotenuse

=Perpendicular Side/Hypotenuse Side

= a/h

Hence, the ratio for sin 60 degrees function is, sin 60 = Perpendicular/Hypotenuse

There's a straightforward method to calculate the value of sine ratios for all degrees. Once you grasp this method, you can easily calculate the values for all other trigonometry ratios. Let's start with the values for sin 0°, sin 30°, sin 45°, sin 60°, sin 90°.

From the above calculations, we can see that the exact value of sin 60 degrees is √3/2. Similarly, we can find the values for cos and tan ratios.

Therefore, the precise value of sin 60 degrees is √3/2

Cos 0° = Sin 90° = 1

Cos 30°= Sin 60° = √3/2

Cos 45° = Sin 45° = 1/√2

Cos 60° = Sin 30° =1/2

Cos 90° = Sin 0° = 0

Also,

Tan 0° = Sin 0°/Cos 0° = 0

Tan 30° = Sin 30°/Cos 30° =1/√3

Tan 45° = Sin 45°/Cos 45° = 1

Tan 60° = Sin 60°/Cos 60° = √3

Tan 90° = Sin 90°/Cos 90°= ∞

The above values of trigonometry ratios are expressed in degrees. However, these values can also be expressed in radians, which is commonly used for a unit circle with a radius of one. The radian is symbolized by π.

For instance, the radian value for 0° is 0. Similarly, we can create a table for trigonometry ratios with respect to π.

Radian	0	π/6	π/4	π/3	π/2	π	3π/2	2π
Sin	0	1/2			1	0	-1	0
Cos	1			1/2	0	-1	0	1
Tan	0		1		Undefined	0	Undefined	0

The value of sin 60° is positive because 60 degrees lies in the first quadrant of the coordinate plane, where all trigonometric functions are positive.
The exact value of sin 60° is √3/2, which is approximately 0.866.

How to Find sin 60°?

1. Using Trigonometric Formulas:
   You can calculate sin 60° using different trigonometric identities:

• sin 60° = √(1 – cos² 60°)
• sin 60° = tan 60° ÷ √(1 + tan² 60°)
• sin 60° = 1 ÷ √(1 + cot² 60°)
• sin 60° = √(sec² 60° – 1) ÷ sec 60°
• sin 60° = 1 ÷ cosec 60°

Since 60° is in the first quadrant, all results will be positive.

2. Using Trigonometric Identities:
   Sin 60° can also be shown using standard angle relationships:

• sin(180° – 60°) = sin 120°
• –sin(180° + 60°) = –sin 240°
• cos(90° – 60°) = cos 30°
• –cos(90° + 60°) = –cos 150°

These help when changing angles or solving equations.

3. Using the Unit Circle:
   You can also find sin 60° using the unit circle method.
   Rotate a line from the center of the circle to make a 60° angle with the positive x-axis. The y-coordinate of that point gives the value of sin 60°, which is √3/2 or 0.866.

Why is the Value of Sin 60° Equal to Sin 120°?

The value of sin 60° is the same as the value of sin 120°, and here's why:

In trigonometry, the sine function is positive in both the first and second quadrants of a circle.
Since both 60° (first quadrant) and 120° (second quadrant) are in areas where sine is positive, their sine values are equal.

We use the identity: sin(180° – θ) = sin(θ)

So, sin(120°) = sin(180° – 60°) = sin(60°)

That means: sin 120° = sin 60° = √3/2, which is approximately 0.866.

Basic Trigonometric Values

Angle (°)	0°	30°	45°	60°	90°	180°	270°
Radians	0	π/6	π/4	π/3	π/2	π	3π/2
sin	0	1/2	1/√2	√3/2	1	0	-1
cos	1	√3/2	1/√2	1/2	0	-1	0
tan	0	1/√3	1	√3	Not Defined	0	Not Defined
cot	Not Defined	√3	1	1/√3	0	Not Defined	0
cosec	Not Defined	2	√2	2/√3	1	Not Defined	–1
sec	1	2/√3	√2	2	–	–1	Not Defined

Examples Using Sin 60 Degrees

Example 1: Find the value of 2 × (sin 45° × cos 45°)
Solution:
We use the identity:
2 × sin A × cos A = sin(2A)

Here, A = 45°
⇒ 2 × sin 45° × cos 45° = sin(2 × 45°) = sin 90°
We know: sin 90° = 1
Answer: 2 × (sin 45° × cos 45°) = 1

Example 2: Simplify: 2 × (sin 30° ÷ sin 390°)
Solution:
Note that sin 390° = sin(360° + 30°) = sin 30°
So, the expression becomes:
2 × (sin 30° ÷ sin 30°) = 2 × (1) = 2
Answer: 2

Example 3: Find the value of sin 45° if cosec 45° is 1.4142
Solution:
We know: sin A = 1 ÷ cosec A
So, sin 45° = 1 ÷ 1.4142 ≈ 0.707
Answer: sin 45° ≈ 0.707

Up until now, we have explored the value of sin 60 degrees along with other degree values. We have also calculated the values for cos and tan degrees in terms of sin degrees and radians. In a similar manner, you can find values for other trigonometric ratios such as sec, cosec, and cot.

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