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 as 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.

For angles less than a right angle, trigonometric functions are commonly defined as the ratio of two sides of a right triangle. The angles are calculated with respect to sin, cos and tan functions. Usually, the degrees are considered as 0°, 30°, 45°, 60°, 90°, 180°, 270° and 360°. Here, we will discuss the value for sin 30 degrees and how to derive the sin 30 value using other degrees or radians.

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:

• 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:

\(\sin \theta =\frac{opposite ~ side}{hypotenuse ~ side}\)

Sine Law: The sine law states that the sides of a triangle are proportional to the sine of the opposite angles.

\(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\)

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 with respect to 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

Derivation to Find the Sin 30 value (Geometrically)

Let us now calculate the sin 30 value. Consider an equilateral triangle ABC. Since each angle in an equilateral triangle is 60°, therefore \(\angle A=\angle B=\angle C=60^{\circ}\)

Draw the perpendicular line AD from A to the side BC (From figure)

Now \(\Delta ABD\cong \Delta ACD\)

Therefore BD=DC and also

\(\angle BAD=\angle CAD\)

Now observe that the triangle ABD is a right triangle, right-angled at D with \(\angle BAD=30^{\circ}\) and \(\angle ABD=60^{\circ}\).

As you know, for finding the trigonometric ratios, we need to know the lengths of the sides of the triangle. So, let us suppose that AB=2a

\(BD=\frac{1}{2}BC=a\)

To find the sin 30-degree value, let’s use sin 30-degree formula and it is written as:

Sin 30° = opposite side/hypotenuse side

We know that, Sin 30° = BD/AB = a/2a = 1 / 2

Therefore, Sin 30 degree equals to the fractional value of 1/ 2.

Sin 30° = 1 / 2

Therefore, sin 30 value is 1/2

In the same way, we can derive other values of sin degrees like 0°, 30°, 45°, 60°, 90°,180°, 270° and 360°. Below is the trigonometry table, which defines all the values of sine along with other trigonometric ratios.

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

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

In the same way,

Sin 0 = sin 180

Trigonometry Table

Solved Examples

Question 1: In triangle ABC, right-angled at B, AB = 5 cm and angle ACB = 30°. Determine the length of the side AC.

Solution:

To find the length of the side AC, we consider the sine function, and the formula is given by

Sin 30°= Opposite Side / Hypotenuse side

Sin 30°= AB / AC

Substitute the sin 30 value and AB value,

\(\frac{1}{2}=\frac{5}{AC}\)

Therefore, the length of the hypotenuse side, AC = 10 cm.

Question 2: If a right-angled triangle has a side opposite to an angle A, of 6cm and hypotenuse of 12cm. Then find the value of angle.

Solution: Given, 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.

Question 3: If a right-angled triangle is having adjacent side equal to 10 cm and the measure of 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

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