Exact Value of sin 7½°

How to find the exact value of sin 7½° using the value of cos 15°?

Solution: 

7½° lies in the first quadrant.

Therefore, sin 7½° is positive.

For all values of the angle A we know that, cos (α - β) = cos α cos β + sin α sin β.

Therefore, cos 15° = cos (45° - 30°) 

cos 15° = cos 45° cos 30° + sin 45° sin 30°

           = \(\frac{1}{√2}\)∙\(\frac{√3}{2}\) + \(\frac{1}{√2}\)∙\(\frac{1}{2}\)

           = \(\frac{√3}{2√2}\) + \(\frac{1}{2√2}\)

           = \(\frac{√3 + 1}{2√2}\)

Again for all values of the angle A we know that, cos A = 1 - 2 sin\(^{2}\)\(\frac{A}{2}\)

⇒ 1 - cos A = 2 sin\(^{2}\) \(\frac{A}{2}\)

⇒ 2 sin\(^{2}\) \(\frac{A}{2}\) = 1 - cos A

⇒ 2 sin\(^{2}\) 7½˚ = 1 - cos 15°

⇒ sin\(^{2}\) 7½˚ = \(\frac{1  -  cos 15°}{2}\)

⇒ sin\(^{2}\) 7½˚ = \(\frac{1 - \frac{√3 + 1}{2√2}}{2}\)

⇒ sin\(^{2}\) 7½˚ = \(\frac{2√2 - √3 - 1}{4√2}\)

⇒ sin 7½˚ = \(\sqrt{\frac{4 - √6 - √2}{8}}\), [Since sin 7½° is positive]

⇒ sin 7½˚ = \(\frac{\sqrt{4 - √6 - √2}}{2√2}\)

Therefore, sin 7½˚ = \(\frac{\sqrt{4 - √6 - √2}}{2√2}\)



● Submultiple Angles






11 and 12 Grade Math

From Exact Value of sin 7 and Half Degree to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.

Share this page: What’s this?

Recent Articles

RSS

  1. Area of a Square | Area of a Square = (Side × Side) Square Units

    Apr 28, 24 01:28 PM

    Square
    In area of a square we will learn how to find the area by counting squares. To find the area of a region of a closed plane figure, we draw the figure on a centimeter squared paper and then count the n…

    Read More

  2. Area of a Rectangle | Area of Rectangle = Length × Breadth | Examples

    Apr 28, 24 12:56 PM

    Area of Regular Figures
    Area of a rectangle is discussed here. We know, that a rectangle has length and breadth. Let us look at the rectangle given below. Each rectangle is made of squares. The side of each square is 1 cm lo…

    Read More

  3. Area | Units to find Area | Conversion Table of Area | Two Dimensional

    Apr 28, 24 10:14 AM

    Concept of Area
    The amount of surface that a plane figure covers is called its area. It’s unit is square centimeters or square meters etc. A rectangle, a square, a triangle and a circle are all examples of closed pla…

    Read More

  4. Math Only Math | Learn Math Step-by-Step | Worksheet | Videos | Games

    Apr 27, 24 02:23 PM

    Presenting math-only-math to kids, students and children. Mathematical ideas have been explained in the simplest possible way. Here you will have plenty of math help and lots of fun while learning.

    Read More

  5. Worksheet on Use of Decimal | Free Printable Decimals Worksheets

    Apr 27, 24 01:45 PM

    Practice the questions given in the worksheet on use of decimals in calculating money, in measuring the length, in measuring the distance, in measuring the mass and in measuring the capacity.

    Read More