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Applications of Trigonometry

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Trigonometry is the branch of mathematics which deals with Triangles and the relationship between the angles and sides of triangles.
Now we will see some applications of Trigonometry:
Some of the applications of trigonometry are given below:
Sine law – The law of sine (is also known as sine law, sine formula, or sine rule) is an equation which is used to compare the length of the sides of a triangle to its angle:

According to the sine law,

$\frac{sin X}{x}$ = $\frac{sin Y}{y}$ = $\frac{sin Z}{z}$
,
where x, y, and z are the length of the sides of a triangle, and X, Y, and Z are the opposite angles of a triangle.
And the reciprocal of above equation is:

$\frac{x}{sin X}$ = $\frac{y}{sin Y}$ = $\frac{z}{sin Z}$

Another application of trigonometry is:

Law of cosines: It is also said to be cosine formula or cosine rule. It is used to compare the lengths of the sides of a triangle to the cosine of one of its angle.

In the mathematical notation the cosine laws says that:

Z2 = x2 + y2 – 2xy cos ∂;

‘∂’ represents the angle between sides of lengths ‘x’ and ‘y’ and the opposite side length ‘z’.

Some other trigonometry applications are:

Suppose in a triangle if we know the two angles and one side of a triangle then SAA or ASA or laws of sine can be used to solve the triangle.
Now we will see how to find one side of a triangle:

Suppose we have a triangle XYZ and if X = 33 degree and ‘Y’ is 82.8 degree and ‘x’ is 43.9 inch. Then we find the value of ‘y’.

We know that the law of sine is:

$\frac{sin X}{x}$ = $\frac{sin Y}{y}$

Putting the values in the formula, we get:

$\frac{sin (33)^{0}}{43.9}$ = $\frac{sin(82-8)^{0}}{y}$

or

Y = 80.05

The Law of Sines to Solve Triangles

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In mathematics, we study about Triangles, relationships between their sides and the angles between these sides; all these come under trigonometric. Among these Trigonometric Functions the sine function comes first in all. It gives the Ratio of the length of the side opposite to an angle (also called as perpendicular) to the length of the hypotenuse.
Mathematically,
Sin (theta) = opposite / hypotenuse
The law of sines also called sine rule or sine law is very useful when we compute the unknown parameters (sides and angles) of the given triangle.
According to the law of sine:
In the triangle with sides a, b, c and the angles A, B, C. If we divide a side ‘a’ by ‘sine A’ it is equal to side ‘b’ divided by the sine of angle ‘B’ and also equal to side ‘c’ divided by ‘sine C’.
Mathematically,
a / sin A = b / sin B = c / sin C,
We can also state it as the reciprocal of the above expression that is:
sin A / a = sin B / b = sin C / c
Let us take a triangle with two angles 350 and 1050 and a side of measure 5 units and we will find unknown side ‘c’.
According to the law of sine:
a / sin A = b / sin B = c / sin C,
Put the given values in the above expression:
a / sin A = 5/ sin (350) = c /sin (1050)
=> 5 / sin(350) = c / sin(1050)
Solving the above equation we have,
c = ( 5 / sin(35°) ) × sin(105°)
=> c = ( 5 / 0.574 ) × 0.966
=> c = 8.041unit
Similarly, we can find any desired measure of angle if the required information is available.

The Law of Cosines

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In mathematics, when we study about Triangles and the relationships between their sides and angles between these sides we use trigonometric Functions. Cosine function gives the Ratio of the length of side adjacent to an angle (also called as base) to the length of the hypotenuse.
Mathematically,
cos (theta) = adjacent / hypotenuse
The Law of Cosines also called Cosine Law and it is very useful for determining the unknown parameters (sides and angles) of any triangle.
According to law of cosine:
a2 = b2 + c2 – 2bc cos A,
b2 = a2 + c2 – 2ac cos B,
c2 = a2 + b2 – 2ab cos A,

Solving these above equations we get,
Cos A = (-a2 + b2 + c2 )/ 2bc
Cos B = (a2 - b2 + c2 )/ 2ac
Cos C = (a2 + b2 - c2 )/ 2ab
If in the above figure we have given the measure of a = 13 and b = 20 with angle A = 660 we can find the length of side “c” using law of cosine formula. For that we can use the above stated cosine rule.
c2 = a2 + b2 – 2ab cos A (as the law of cosine formula)
Now substitute the value of ‘side a’ and ‘side b’ we have c2 = 13² + 20² -2* 20 * 13 * cos (66)
c2 = 358
c = √358
c = 18.9
Let us take prove of this law:
Considering the above triangle with sides a, b, c, where “C” is the measurement of the angle opposite the side of length c.
Suppose the triangle is placed in coordinate system:
a = (b cos C, b sin C)
b = (a, 0),
c = (0, 0),
By the distance formula, we have:
C = √[(a – b cos C)2 + (0 – b sin C)2],
=> c2 = (a – b cos C)2 + (0 – b sin C)2,
=> c2 = a2 – 2abcosC + b2cos2C + + b2sin2C,
=> c2 = a2 + b2 – 2ab cos C.

De Moivre's Theorem

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De moivre’s theorem is used to calculate the roots of a complex for any power ‘n’, and the value of ‘n’ is Integer. As we know de moivres theorem is obtained from Euler’s equation. It is also used to join the Trigonometry to the Complex Number. The formula for demoivre s theorem is given by:
⇒ (cos p + isin p)n = cos (np) + I sin (np). This formula is used to join the complex number and trigonometry. Here ‘I’ denotes an Imaginary Number. The value of iota Square is ‘-1’ (i2 = -1). If we expand the left hand side, of the above formula then we have to compare real and imaginary parts and it is also possible to solve the expression cos np and sin np in terms of sin p and cos p.
Now, we will see how we find derivative of De moivre’s theorem? As we know, De moivre’s theorem is obtained from Euler’s formula:
So we can write it as:
⇒ eip = cos p + I sin p, and the exponential law for integer powers is:
⇒ (eip)n = einx;
Then the Euler’s formula is:
⇒ ei (np) = cos (np) + isin (np);
In general, De Moivre’s theorem does not hold for non – integer powers. The above de moivre’s theorem does not include complex number to the power n.
Suppose z1 and z2 be two complex Numbers, where the value of | z1| = r1, | z1| = r1, and | z2| = r2, and arg (z1) = ⊖1, arg (z1) = ⊖2.
Then, z1 = r1 (cos ⊖1 + i sin ⊖1);
And z2 = r2(cos ⊖2 + i sin ⊖2);
This is all about De moivre’s theorem.