Sin 45 Degree

Sin 45 Degrees

The value of Sin 45 degree in decimal form is 0.7071067812. 

Sine is considered as one of the most important functions in trigonometry as it is used to find out the unknown values of the angles and length of the sides of a right-angle triangle. It is necessary to talk about the importance of Sine functions in trigonometry before having a discussion on Sin 45 degrees. In the right-angle triangle, Sine function can be defined as the ratio of the length of the opposite side of a triangle to its hypotenuse. We generally require the value of Sin 45 degree and the value of other trigonometry ratios for all the degrees such as 0, 30°,45°,60°,90°,180° to resolve the questions based on trigonometry.

We will discuss Sin 45 value in this article.

Sin 45 Value

Let us take a right angle triangle Δ ABC to know about the Sin 45 value. The Sin ϴ of an angle can be defined as the ratio of the length of the opposite side of a triangle to its hypotenuse.

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Based on the above diagram:

Sin ϴ = Opposite side or perpendicular / Hypotenuse

Sin ϴ = AC/AB

Sin ϴ  =  b/c

We will discuss here a simple procedure through which you can easily find out the Sin 45 value and the value of other Sin ratios with different degrees. You will be able to calculate values of all the trigonometry ratios which are used to resolve trigonometry equations if you understand this method thoroughly.

Calculation of the Value of Sin 45 Degree in Fraction

  • Sin 0° = √0 /√4 = 0

  • Sin 30° = √1 /√4 =1/2

  • Sin 45° = √2 /√4 = 1/√2

  • Sin 60° = √3 /√4 = √3 /2

  • Sin 90° = √4 /√4 = 1

Similarly, the value of the other trigonometry ratios such as cosine and tangent can be found out.

Now, we will learn sine ratios for the different degree based on the table given below:

Sine Ratio Table

Sin 0°

0

Sin 30°

1/2

Sin 45°

1/√2

Sin 60°

√3/2

Sin 90°

1

Sin 120°

√3/ 2

Sin 150°

1/2

Sin 180°

-1

Sin 270°

0


What is the Exact Value of Sin 45° Geometrically?

Let us find the exact value of Sin 45 using an isosceles triangle

An isosceles triangle with perpendicular side =1 cm and hypotenuse = √2 will give you the value of Sin 45 degree.

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As we know,

Sin 45° = Perpendicular/ hypotenuse

Sin 45 ° = AC/ABC

Sin 45° =        1/√2

So, the exact value of Sin 45 degree is 1/√2

The exact value of Sin 45 degree in decimal form is 0.7071067812

Here are Some Sine Formulas According to Compound Angles:

Sin (α+ B) = Sin α + Cos β+ Cos β + Sin β

Sin (α - β) = Sin α - Cos β- Cos β - Sin β

 

Solved Examples

Evaluate the Following

a. 2 Sin 30° + 3 Cos 60° - 3 Tan 45°

   Solution – Value of Sin 30° = ½

    Value of Cos 60° = ½

    Value of tan 45° = 1

Substituting the Values,

= 2(1/2) + 3(1/2)- 3(1)

= 1+ 3/2 – 3

= - 1/2

b.  3( Cos 30°)² + 2( Sin 30°)²

Solution- Value of Cos 30° = √3/2

                Value of Sin °30 = ½

Substituting the values,

=3(√3/2)² +2(1/2)²

=3(3/4) + 2(1/4)

= 9/4 + 2/4

= 11/4

c.     Sin (45°+30)=  √2( √3+1) / 4

 Sin (α+ B) = Sin α + Cos β+ Cos β + Sin β

Solution: Substituting the values using the above formula, we get 

Sin (45°+30) = Sin 45° Cos 30° + Cos 45° Sin 30°

= Sin (45°+30° ) = √2/2 × √3/2 +√2/2 ×1/2

Taking √2 as common, we get

 Sin (45°+30° ) = √2 (√3 + 1) / 4

Hence Proved

Fun Fact

  • The 8 trigonometric identities used in trigonometry are also known as fundamental identities. Among 8 trigonometry identities, 3 of the identities are known as Pythagoras identities as they are based on the Pythagorean Theorem.

  • Babylonians, the Egyptians based Trigonometry on their base 60 numeral system.

  • Trigonometry is primarily associated with architecture and music.

  • The two most important trigonometry functions which are widely used are Sine and Cosine.

Quiz Time

1. The two trigonometry functions such as Sine and Cosine are different by the addition of the prefix "co" to "cosine". From where does the word "co" derive?

a.       Complementary

b.      Constraint

c.       Coefficient

d.      Constant

2. The sine of an angle ϴ is equivalent to

a.       Opposite side divided by adjacent side

b.      Opposite side divided by hypotenuse

c.       Adjacent side divided by Hypotenuse

d.      Hypotenuse divided by the adjacent

FAQ (Frequently Asked Questions)

1. Explain Trigonometry Ratios.

Six different trigonometry ratios are used to solve trigonometry equations. These ratios are sine, cosine, tangent, secant, cosecant, and cotangent. The ratio of the length of the sides of a right-angle triangle in terms of any acute angles known as the trigonometry ratio of that specific angle. The trigonometry ratio applied in the right angle triangle is used to find out the unknown angle and length of the sides of a right-angle triangle.

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Concerning ∠ C in above Δ ABC, the ratio of Sin 45 will be equal to:

Sin 45 = perpendicular/Hypotenuse

Sin 45 = AC / BC

Sin 45 = a /√2 a

2. What are the Properties of Sine Functions According to the Quadrants?

The values of the Sin function can be either positive or negative. These values are dependent upon the quadrants. In the below quadrant table you can see, the value of Sin 90 is positive while the value of Sine 270 is negative. These values depended upon the quadrants in which they lie. If the particular Sine values you want to know lies in the positive quadrant, then its value will be positive. But, that Sine value will be considered as negative if it lies in the negative quadrant.

Positive/Negative Values in 4 Quadrants

1st Quadrant =positive

2nd Quadrant =Negative

3rd Quadrant = Positive

4th Quadrant = Negative

Four Quadrants in Trigonometry