Sin 30 Degrees- Complete Details

The value of sin 30 degrees is 0.5. Sin 30 is also written as sin π/6, in radians. The trigonometric function also called an angle function, relates the angles of a triangle to the length of its sides. Trigonometric functions are important in the study of periodic phenomena like sound and light waves, average temperature variations and the position and velocity of harmonic oscillators, and many other applications. The most familiar three trigonometric ratios are sine function, cosine function, and tangent function. Read other details also provided by Edureify on Sin 30 degrees.

What is the Sine Function?

The sine function is one of the primary trigonometric functions. It relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. The sine function can be defined using the unit circle, where:

• Sine (sin): For an angle $\theta$, $\sin \theta$ represents the y-coordinate of the point on the unit circle that corresponds to that angle.

The Importance of Sine in Trigonometry

Sine is a fundamental function used in various fields, including physics, engineering, and computer graphics. Understanding sine values for different angles is crucial for solving problems involving waves, oscillations, and rotational motion.

Derivation of Sin 30 Degrees

To derive the value of sin⁡30∘\sin 30^\circsin30∘, we can use the unit circle or the properties of a right triangle.

1. Using the Unit Circle:
   • In the unit circle, the angle of 30∘30^\circ30∘ corresponds to a point with coordinates (32,12)(\frac{\sqrt{3}}{2}, \frac{1}{2})(23​​,21​).
   • Here, the y-coordinate represents sin⁡30∘\sin 30^\circsin30∘, confirming that sin⁡30∘=12\sin 30^\circ = \frac{1}{2}sin30∘=21​.
2. Using a Right Triangle:
   • Consider a 30-60-90 triangle, where the ratio of the sides is 1:3:21 : \sqrt{3} : 21:3​:2.
   • In this triangle, the side opposite the 30∘30^\circ30∘ angle is $1$ (half of the hypotenuse), leading to the same conclusion:

   sin⁡30∘=oppositehypotenuse=12.\sin 30^\circ = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}.sin30∘=hypotenuseopposite​=21​.

Real-World Applications of Sine

Understanding the value of sin⁡30∘\sin 30^\circsin30∘ is not just academic; it has real-world applications:

• Physics: Sine functions are used to describe wave patterns, such as sound waves and light waves.
• Engineering: Engineers use sine in the design of structures and systems, especially those involving angles and slopes.
• Computer Graphics: Sine functions help simulate movements and animations in video games and simulations.

Sine 30 Degrees Value

The exact value of sin 30 degrees is ½. To define the sine function of an angle, start with a right-angled triangle ABC with the angle of interest and the sides of a triangle. The three sides of the triangle are given as follows:

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• The opposite side is the side opposite to the angle of interest.
• The hypotenuse side is the side opposite the right angle and it is always the longest side of a right triangle
• The adjacent side is the side adjacent to the angle of interest other than the right angle
• The sine function of an angle is equal to the length of the opposite side divided by the length of the hypotenuse side and the formula is given by:

 

• Sin 30:- Opposite side/Hypotenuse Side

 

The sine rule is used in the following cases :

 

Case 1: Given two angles and one side (AAS and ASA)

Case 2: Given two sides and non included angle (SSA)

The other important sine values concerning angle in a right-angled triangle are:

Sin 0 = 0

Sin 45 = 1/√2

Sin 60 = √3/2

Sin 90 = 1

 

Fact: The values sin 30 and cos 60 are equal.

Sin 30 = Cos 60 = ½

And

Cosec 30 = 1/Sin 30

Cosec 30 = 1/(½)

Cosec 30 = 2

Why Sin 30 is equal to Sin 150

The value of sin 30 degrees and sin 150 degrees are equal.

Sin 30 = sin 150 = ½

Both are equal because the reference angle for 150 is equal to 30 for the triangle formed in the unit circle. The reference angle is formed when the perpendicular is dropped from the unit circle to the x-axis, which forms a right triangle.

Since the angle of 150 degrees lies on the IInd quadrant, therefore the value of sin 150 is positive. The internal angle of a triangle is 180 – 150=30, which is the reference angle.

The value of sine in the other two quadrants, i.e. 3rd and 4th are negative.

In the same way,

Sin 0 = sin 180

Some Solved Questions on Sin 30

 

Question 1: If a right-angled triangle has a side opposite to an angle A, of 6cm and hypotenuse of 12cm. Then find the value of the angle.

 

Solution: Given, the Side opposite to angle A = 6cm

Hypotenuse = 12cm

By sin formula we know that;

Sin A = Opposite side to angle A/Hypotenuse

Sin A = 6/12 = ½

We know, Sin 30 = ½

So if we compare,

Sin A = Sin 30

A = 30

Hence, the required angle is 30 degrees.

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Question 2: If a right-angled triangle is having adjacent side equal to 10 cm and the measure of an angle is 45 degrees. Then find the value hypotenuse of the triangle.

 

Solution: Given adjacent side = 10cm

We know,

Tan 45 = Opposite side/Adjacent side

Tan 45 = opposite side/10

Since Tan 45 = 1

Therefore,

1 = Opposite side/10

Opposite side = 10 cm

Now, by sin formula, we know,

Sin A = Opposite side/Hypotenuse

So,

Sin 45 = 10/Hypotenuse

Hypotenuse = 10/sin 45

Hypotenuse = 10/(1/√2)

Hypotenuse = 10√2

 

Frequently Asked Questions – FAQs

 

Question No 1:- What is the exact value of sin 30 degrees?

Ans:- The exact value of sin 30 degrees is 0.5.

 

Question No 2:- What is the value of sin 30 in the form of a fraction?

Ans:- The value of sin 30 in the form fraction is ½.

Question no 3:-What is sine 30 degrees in radian?

Ans:- Sine 30 degrees in radian is given by sin π/6.

Question No 4:- What is the formula for sine function?

Answer:- Sine of an angle, in a right-angled triangle, is equal to the ratio of opposite sides of the angle and hypotenuse.

Sine A = Opposite Side/Hypotenuse

 

Question No 5:- What is the value of cos 30 and tan 30?

Ans:- The value of cos 30 degrees is √3/2 and the value of tan 30 degrees is 1/√3.

Practice Problems

Now that you understand sin⁡30∘\sin 30^\circsin30∘, test your knowledge with these practice problems:

1. What is the value of sin⁡60∘\sin 60^\circsin60∘?
2. If sin⁡θ=12\sin \theta = \frac{1}{2}sinθ=21​, what are the possible angles $\theta$ in the range 0∘0^\circ0∘ to 360∘360^\circ360∘?
3. Draw a right triangle with a 30∘30^\circ30∘ angle and label the opposite, adjacent, and hypotenuse sides.

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