Introduction
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 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 and the value of other trigonometry ratios for all the degrees such as 0, 30°,45°,60°,90°,180° to solve 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. If you understand this method thoroughly, you will be able to calculate the values of all the trigonometric ratios which are used to solve trigonometry equations.
Note:
Opposite side: It's the side facing the angle we're interested in, which is angle B in our case.
Hypotenuse: The longest side of the right triangle, located opposite the right angle.
Calculation of the Value of Sin 45 Degree in Fraction
Sin 0° = \[\frac{\sqrt{0}}{\sqrt{4}}\] = 0
Sin 30° = \[\frac{\sqrt{1}}{\sqrt{4}}\] =1/2
Sin 45° = \[\frac{\sqrt{2}}{\sqrt{4}}\] = \[\frac{1}{\sqrt{2}}\]
Sin 60° = \[\frac{\sqrt{3}}{\sqrt{4}}\] = \[\frac{\sqrt{3}}{2}\]
Sin 90° = \[\frac{\sqrt{4}}{\sqrt{4}}\] = 1
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
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 and hypotenuse = √2 will give you the value of Sin 45 degrees.
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As we know,
Sin 45° = Perpendicular/ hypotenuse
Sin 45 ° = AB/AC
Sin 45° = \[\frac{1}{\sqrt{2}}\]
So, the exact value of Sin 45 degree is \[\frac{1}{\sqrt{2}}\]
The exact value of Sin 45 degree in decimal form is 0.7071067812.
Here are some Sine Formulas according to Compound Angles:
\[Sin ( \alpha + \beta ) = Sin \alpha + Cos \beta + Cos \beta + Sin \beta \]
\[Sin ( \alpha - \beta ) = Sin \alpha - Cos \beta - Cos \beta - Sin \beta \]
Importance of Sine and Other Trigonometric Functions
In the general population of non-mathematicians and 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 and 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 and geodesy, many physical sciences, mechanical engineering, machining, medical imaging, number theory, oceanography, optics, pharmacology, probability theory, seismology, statistics and visual perception.
Applications of Sin 45 Degrees
The applications of Sin 45 degrees are as diverse as they are fascinating. Here are a few examples:
Solving right triangles: Knowing sin 45 degrees allows you to find missing side lengths and angles in 45-45-90 triangles using other trigonometric ratios like cosine and tangent.
Example: Isosceles Right Triangles, In a 45°-45°-90° isosceles right triangle, each leg is √2/2 times the hypotenuse, a direct consequence of Sin 45°.
Building bridges and structures: Engineers rely on trigonometry to calculate forces and angles in structures, and sin 45 degrees plays a crucial role in ensuring stability and safety.
Understanding projectile motion: From launching rockets to predicting the trajectory of a thrown ball, sin 45 degrees helps us understand how objects move under the influence of gravity.
Animating graphics and games: Computer graphics rely heavily on trigonometry to create realistic movements and shapes, and sin 45 degrees is often used to define rotations and transformations.
Harmonizing music: The ratios between musical notes can be expressed using trigonometric functions, and sin 45 degrees forms the basis for certain chords and harmonies.
Solved Examples
Evaluate the following:
2 Sin 30° + 3 Cos 60° - 3 Tan 45°
Ans: Value of Sin 30° =\[\frac{1}{2}\]
Value of Cos 60° = \[\frac{1}{2}\]
Value of tan 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° = \[\frac{\sqrt{3}}{2}\]
Value of Sin °30 = \[\frac{1}{2}\]
Substituting the values,
=\[3(\frac{\sqrt{3}}{2})^{2} + 2(\frac{1}{2})^{2}\]
=3(3/4) + 2(1/4)
= 9/4 + 2/4
= 11/4
Sin (45°+30)= \[\frac{\sqrt{2}(\sqrt{3}+1)}{4}\]
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° ) = \[\frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2}\]
Taking \[\sqrt{2}\] as common, we get
Sin (45°+30° ) = \[\frac{\sqrt{2}(\sqrt{3}+1)}{4}\]
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 trigonometric functions which are widely used are Sine and Cosine.
Quiz Time
The two trigonometric functions such as Sine and Cosine are different by the addition of the prefix "co" to "cosine". From where does the word "co" derive?
Complementary
Constraint
Coefficient
Constant
The sine of an angle ϴ is equivalent to
Opposite side divided by the adjacent side
The opposite side is divided by the hypotenuse
Adjacent side divided by the hypotenuse
Hypotenuse divided by the adjacent side
Conclusion
By connecting sin(45°) with cosine and tangent values, we uncover interesting relationships between trigonometric functions. Graphing the sine function at different angles, especially around 45°, gives us a visual understanding of how sine values behave. Beyond just numbers, sin(45°) finds practical use in fields like distance calculations, angle determinations, and force assessments. So, delving into sin(45°) isn't just about numbers; it's an exciting journey into the world of math and its real-world applications.
FAQs on Sin 45 Degree
1. Explain trigonometric ratios.
Six different trigonometric 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 angle is known as the trigonometry ratio of that specific angle. The trigonometric ratio applied in a 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:
\[Sin 45 = \frac{Perpendicular}{Hypotenuse}\]
\[Sin 45 = \frac{AC }{BC}\]
\[Sin 45 = \frac{a}{\sqrt{2}a}\]
2. What are the properties of sine functions according to the quadrants?
If we plot the values of various sine functions on a graph, the point when trailed gives rise to a wave-like symmetry. There are a total of five major points that are plotted (sin 0, sin 30, sin 45, sin 60, and sin 90). The value of the sine function is maximum for sin 30 and sin 60, albeit in the complementary direction of the Y-axis. Similarly, the values of sin 0, sin 45, and sin 90 are located equidistantly on the Y-axis.
The points follow further as periodic functions which have a period= \[2\pi \]
The domain of each function of sine ranges from -1 to +1
The graph of y= six shows symmetry around its origin and is an odd trigonometric function.
3. What are the other commonly used sine values in trigonometry? How are they useful?
The other commonly used sine values in trigonometry are sin15, sin30, sin60, and sin 90. The value of these sine values are:
Sin 0 = 0
Sin 30 = 1/2
Sin 60 = \[\frac{\sqrt{3}}{2}\]
Sin 90 = 1
All other sine values can be derived as multiples of the above sine functions. In higher physics and mathematics, one must understand this important yet simple concept to quickly calculate and solve mathematical problems and numericals.
4. What is the use of the trigonometric ratio: sin 45?
Sine 45 is an important trigonometric ratio as explained in this article.
It is used to solve mathematical problems in both maths and physics.
It is used for architectural purposes (constructing bridges, buildings, etc).
It is used to create various devices, etc.
It is used in the understanding of concepts such as sound waves, electric current, etc.
5. Does sin 45 and cos 45 refer to the same thing?
No, sin 45 and cos 45 do not refer to the same trigonometric function since the trigonometric ratios of sine and cosine are derived differently from a right-angled triangle. However, when solved, they give the same answer (or mathematical value of \[\frac{1}{\sqrt{2}}\]). From the circle, we can understand that the function of cosine is simply a complementary trigonometric ratio to the sine. Hence, the complement of cosine will come out to be the same as sine at one point. This is the 45th degree for both of them.