Sine is considered 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-angled triangle. It is necessary to lớn talk about the importance of Sine functions in trigonometry before having a discussion on Sin 45 degrees. In a right-angled triangle, the 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 degrees & the value of other trigonometry ratios for all the degrees such as 0, 30°,45°,60°,90°,180° lớn solve the questions based on trigonometry.
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We will discuss Sin 45 value in this article.
Importance of Sine và Other Trigonometric Functions
In the general population of non-mathematicians & non-scientists, trigonometry is mainly thought to be useful in measurement problems. But many remain unaware of how trigonometry silently seeps into their daily lives as well. For example, in the theory of music.
In other cases, the use is still more technical, such as in the number theory. Many mathematical concepts such as the Fourier series và Fourier transforms are heavily dependent on the knowledge of trigonometric functions that find application in a number of areas including statistics.
Traditionally, many scientific fields mentioned make use of trigonometry. They are acoustics, architecture, astronomy, cartography, civil engineering, geophysics, crystallography, electrical engineering, electronics, land surveying & geodesy, many physical sciences, mechanical engineering, machining, medical imaging, number theory, oceanography, optics, pharmacology, probability theory, seismology, statistics and visual perception.
Great Trigonometrical Survey
The Great Trigonometrical Survey was a massive project undertaken during the years 1802 up until 1871 lớn survey the Indian subcontinent with high precision. Mathematicians & geographers triangulated vast distances across the country starting from the coastal baseline. One of the key achievements of this survey was that for the first time they were able to lớn scale the height of the Himalayan mountains và conclude that Mount Everest is the highest point on the Earth.
One important thing to note is that in some of the classical fields of mathematical sciences such as navigation và land surveying, the use of trigonometry is obvious. However, in other non-classical cases, such as the development of music theory, the application of trigonometry is hidden. Yet some musicians might know how it is related to lớn work begun by Pythagoras, who observed that the sounds made by plucking two strings of different lengths are consonant in the case when both lengths are small integer multiples of a common length. The similarities between the shape of a vibrating string, the shapes of some waves, etc., và the graph of the sine function is not merely coincidental. Rather, these are specific studies originating from the knowledge of periodicity of sine và cosine functions.
Sin 45 Value
Let us take a right angle triangle Δ ABC to lớn 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 khổng lồ its hypotenuse.
(Image Will be uploaded soon)
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. If you understand this method thoroughly, you will be able lớn calculate the values of all the trigonometric ratios which are used to lớn solve trigonometry equations.
Calculation of the Value of Sin 45 Degree in Fraction
Sin 0° =
Sin 30° =
Sin 45° =
Sin 60° =
Sin 90° =
Similarly, the values of other trigonometric ratios such as cosine and tangent can be found out.
Now, we will learn sine ratios for the different degrees based on the table given below.
Sine Ratio Table
Sin 0° | 0 |
Sin 30° | |
Sin 45° | |
Sin 60° | |
Sin 90° | 1 |
Sin 120° | |
Sin 150° | |
Sin 180° | -1 |
Sin 270° | 0 |
Q. What is the exact value of Sin 45° geometrically?
Ans: Let us find the exact value of Sin 45 using an isosceles triangle.
An isosceles triangle with perpendicular side =1 cm & hypotenuse = √2 will give you the value of Sin 45 degrees.
(Image Will be uploaded soon)
As we know,
Sin 45° = Perpendicular/ hypotenuse
Sin 45 ° = AB/AC
Sin 45° =
So, the exact value of Sin 45 degree is
The exact value of Sin 45 degree in decimal form is 0.7071067812.
Here are some Sine Formulas according to lớn Compound Angles:
Solved Examples
Evaluate the following:
2 Sin 30° + 3 Cos 60° - 3 rã 45°
Ans: Value of Sin 30° =
Value of Cos 60° =
Value of chảy 45° = 1
Substituting the Values,
= 2(1/2) + 3(1/2)- 3(1)
= 1+ 3/2 – 3
= - 1/2
3( Cos 30°)<^2> + 2( Sin 30°)<^2>
Ans: Value of Cos 30° =
Value of Sin °30 =
Substituting the values,
=<3(fracsqrt32)^2 + 2(frac12)^2>
=3(3/4) + 2(1/4)
= 9/4 + 2/4
= 11/4
Sin (45°+30)=
Sin (α+ B) = Sin α + Cos β+ Cos β + Sin β
Ans: Substituting the values using the above formula, we get
Sin (45°+30) = Sin 45° Cos 30° + Cos 45° Sin 30°
= Sin (45°+30° ) =
Taking
Sin (45°+30° ) =
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 & music.
The two most important trigonometric functions which are widely used are Sine and Cosine.
Quiz Time
The two trigonometric functions such as Sine & Cosine are different by the addition of the prefix "co" lớn "cosine". From where does the word "co" derive?
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Yi Sun-shin, also spelled Yi Sun-sin, (born April 28, 1545, Seoul, Korea
After passing the government examinations khổng lồ become a military officer in 1576, Yi served at various army and navy posts. Although he was twice discharged after being falsely accused by jealous colleagues, in 1591 he was appointed commander of the naval forces in Left Chŏlla province, where he concentrated on training his men, stocking equipment và supplies, và developing the renowned kŏbuksŏn (“turtle ship”). The kŏbuksŏn is thought to lớn have been the first ironclad battleship in history. Its upper deck was covered with armoured plates khổng lồ protect its crew, and spikes và knives were attached khổng lồ the plates khổng lồ discourage enemies from boarding. The ship’s bow was equipped with a dragon head through which cannon could be fired and clouds of smoke could be emitted to obscure the ship’s position. Cannon và guns could also be fired from the stern & the sides of the ship.
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As a result of Yi’s preparations, his forces, unlike most of the Korean military, were ready to lớn fight when the Japanese invaded in 1592. Yi’s victories off the southern coast effectively cut off the Japanese troops in Korea from supplies và reinforcements & prevented the Japanese from pressing their initial advantage. In 1593 Yi was given command of the entire Korean fleet, but, following peace negotiations, in 1597 he was again falsely accused of disloyalty và demoted to the rank of common soldier. The Japanese then launched a second invasion and succeeded in destroying almost all of the Korean navy. Yi was reinstated as commander of the few remaining ships and, continuing his undefeated battle record, soon restored Korea’s control of the seas. He was killed by a stray bullet as he pursued the retreating Japanese forces during the final chiến dịch of the war.