Value of sin 15

Value of sine 15 degrees can be evaluated easily. The whole trigonometric functions and formulas are designed based on three primary ratios. These ratios are Sine, cosine, and tangent in trigonometry. These ratios help us in finding angles and lengths of sides of a right triangle.

We use sin, cos, and tan functions to calculate the angles. The degrees used commonly are 0, 30, 45, 60, 90, 180, 270 and 360 degrees. We use these degrees to find the value of the other trigonometric angles like the value of sine 15 degrees.

What is the value of Sin 15°?

The actual value of sin 15 degrees is given by:

How to find the Value of Sin 15 Degree?

Method 1: We can find the value of Sin 15° with the help of sin 30 degrees.

(Sin P/2 + Cos P/2)² = Sin² P/2 + Cos² P/2 +2Sin P/2Cos P/2

= 1 + sinP

Sin P/2 + Cos P/2 = ± _√ (1 + sin P)

If P = 30° so P/2 = 30/2 =15°

Putting this value in the above equation:

Sin 15° + Cos 15° = ±_√ (1 + sin 30) …(1)

Also, (Sin P/2 – Cos P/2)² = Sin² P/2 + Cos² P/2 – 2Sin P/2Cos P/2

= 1 – sinP

Sin P/2 – Cos P/2 = ± _√(1 – sin P)

Putting this value in the above equation:

Sin 15° – Cos 15° = ±_√(1 – sin 30°) …(2)

As seen, sin 15° > 0 and cos 15° > 0

hence, sin 15° + cos 15° > 0

From (1) we will get,

sin 15° + cos 15° = √(1 + sin 30°) …(3)

Also, sin 15° – cos 15° = √2 (1/√2 sin 15˚ – 1/√2 cos 15˚)

or, sin 15° – cos 15° = √2 (cos 45° sin 15˚ – sin 45° cos 15°)

or, sin 15° – cos 15° = √2 sin (15˚ – 45˚)

or, sin 15° – cos 15° = √2 sin (- 30˚)

or, sin 15° – cos 15° = -√2 sin 30°

or, sin 15° – cos 15° = -√2 x 1/2

or, sin 15° – cos 15° = – √2/2

So, sin 15° – cos 15° < 0

Now we got, from (2) sin 15° – cos 15°= -√(1 – sin 30°) … (4)

Adding eq. (3) and (4) we get,

2 sin 15° = √(1 + ½) – √(1 – ½)

2 sin 15° = (√3−1)/√2

∴ sin 15° = (√3−1)/2√2

Method 2: Another method to find the value of sin 15 degrees is given below.

Sin 15° =  Sin (45 – 30)°

We know, by trigonometric identities,

Sin (A-B) = Sin A Cos B – Cos A Sin B

Therefore,

Sin (45-30)° = Sin 45° cos 30° – cos 45° sin 30°

Sin 15° = (1/√2)(√3/2)-(1/√2)(1/2)

Sin 15° = (√3/2√2)-(1/2√2)

Sin 15° = (√3–1)/2√2

Value of Sin 15 in Decimal

Since, we know, Sin 15° = (√3–1)/2√2.

Let us rationalise the denominator on the Right-hand side of the above expression.

⇒   (√3–1)/2√2

Multiply and divide √2.

⇒ (√3–1)/2√2 x (√2/√2)

⇒ [√2(√3-1)]/4

⇒ (√6 – √2)/4

⇒ 0.2588 (up to four decimal places)

Value of Cos 15°

If we the value of sin 15 degrees, then we can easily evaluate the value of cos 15 degrees, by using trigonometry identity.

sin² a + cos² a = 1

Put a = 15°

Sin²15° + cos² 15° = 1

Since, Sin 15° = (√3–1)/2√2

[(√3–1)/2√2]2 + cos215 = 1

On solving the above equation, we get;

Cos 15° = √3+1/2√2

Solved Examples

Q.1: Find the value of sin 15° – cos 15°?

Solution: Since, we know,

Sin 15° = (√3–1)/2√2

Cos 15° = √3+1/2√2

Sin 15° – cos 15° = [(√3–1)/2√2] – [√3+1/2√2]

= 1/2√2 [√3–1 – √3–1]

= 1/2√2 [-2]

= -1/√2

Q.2: Find the value of sin 60° + Sin 15°?

Solution: We know that,

Sin 60° = √3/2

Sin 15° = (√3–1)/2√2

So on adding both the values, we get;

Sin 60° + Sin 15° = √3/2 + [(√3–1)/2√2]

= [2√3 + √6 – √2]/4

Register at BYJU’S to learn all about trigonometry and its formulas and download BYJU’S-The Learning App to study with the help of video lessons.