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Right Triangles Trigonometry

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In a Right Triangle the side opposite to the Right Angle is called the hypotenuse and side opposite to the angle is opposite side and other side is adjacent side.


Fig:- Right triangle, right angled at ‘C’.

Trigonometry is a branch of mathematics in which we study Triangles and relationships between their sides and the angles between these sides.
The three primary Functions used in right triangle Trigonometry are:
1. Sine abbreviated as sin.
2. Cosine abbreviated as cos.
3. Tangent abbreviated as tan.

In Right Triangles Trigonometry, the value for the sine of angle ‘A’ is defined as the value that we get after dividing opposite side by the hypotenuse, i.e.

Sin A = opposite side / hypotenuse
= a/c

The value for the cosine of angle ‘A’ is defined as the value that we get after dividing adjacent side by the hypotenuse, i.e.

Cos A = adjacent side / hypotenuse,
= b/c,
The value for the Tangent of angle ‘A’ is defined as the value that we get after dividing the opposite side by the adjacent side, i.e.
Tan A = opposite side / adjacent.
= a/b.
To remember the formulas, we can learn the following statement
“oh heck, another hour of Algebra!”
Here,
o = opposite side
h = hypotenuse
a = adjacent side

First letter represent the formula of the function

oh heck, sin A = o/h,
another hour cos A = a/h,
of algebra tan A = o/a,
Using right triangles and trigonometry we can solve many problems, let's take an example.
Suppose we have to find the length of side “a” in the given right triangle ABC if angle ‘A’ is 35°, and side ‘c’ is 10 inch.

As angle ‘A’ is 36°, then angle B is 90° − 35° = 55°.
To find side we use the formula,
sin A = opposite side / hypotenuse
=> sin 35 = a / 10,
=> 0.5735 = a / 10 (as sin 35= 0.5735),
=> a = 5.735 inch.

The Ratios of Sides

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Ratio is defined as a comparison between any two Numbers of same identities that represents the presence of one entity with respect to other. That means it shows how much quantity of one entity is present in comparison with the other entity. It can also be termed as relationship between two entities or identities of same type. It is represented by “a:b” or “a/b” or “a to b”. Ratios of sides are defined as the Ratio between any two sides of a triangle or comparison or relationship between two sides of triangle. Ratio Triangle Sides is basically defined for Right Angle Triangles, as it is easier to determine the sides in a right angle triangle, by a property known as pythagoras theorem. According to this property, sum of squares of perpendicular and base is equals to Square of hypotenuse. Triangle ratio of sides thus contains the ratio between any two sides of triangle.
Mainly, there are six types of ratio triangle sides in right angle triangle; they are termed as trigonometric ratios. These six terms defines the ratios between different pairs of sides of triangle. As we know, a right angle triangle consists of three sides particularly known as perpendicular, base and hypotenuse. In this different pairs of ratios defines different trigonometric ratios.
First ratio is of perpendicular to hypotenuse known as sine trigonometric ratio.
Second ratio is of base to hypotenuse known as cosine trigonometric ratio.
Third ratio is termed as Tangent ratio which is the comparison of perpendicular and base.
Fourth ratio is termed as cot ratio or cotangent ratio, it is inverse of tangent.
Similarly, cosec and sec are the inverse of sine and cosine respectively.
In mathematical terms these are expressed as:
Sin x= P/H , Cos x= B/H, tan x= P/B, cot x= B/P, sec=H/B and cosec= H/P. here, P is perpendicular, B is Base and H is Hypotenuse that are three sides of triangle and on the basis of these three sides, triangle ratio of sides is defined.

Definition of the Trigonometric Functions of an Acute Angle

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Before going to discuss the definition of the Trigonometric Functions of an Acute Angle let's know about acute angle. An acute angle is an angle which is always lesser than 90 degrees. It does not include 90 degrees anyways. In order to find Trigonometry functions we need an acute angled triangle. And the triangle must be a Right Triangle. The angle which is of 90 degrees is called the Right Angle in a triangle. The opposite side to a right angle is called hypotenuse and rest sides will be called as legs of that triangle.


Here three legs are associated with the angle θ. One is the hypotenuse which is belonging to the angle θ. The second is opposite side and the last is the adjacent side. We denote hypotenuse by r, opposite by b and adjacent side by a.

In a right angle triangle any two sides of right triangle have a Ratio in the form of a relation which is one to one. It helps to form the different Trigonometry formulas and from this we can derive six formulas as the ratio of hypotenuse and opposite, opposite and adjacent, adjacent and hypotenuse and so on.
Now let see the trigonometry function of acute angles θ in the form of ratio of the sides of a right triangle are:
1: sinθ= b/r
2: cosθ= a/r
3: tanθ= b/a
4: cscθ= r/b
5: secθ= r/a
6: cotθ= a/b
Note that sinθ is reciprocal of csc θ, cos θ of sec θ and tan θ of cot θ.
Now see the Pythagorean formula for r, ‘a’ and ‘b’.
Here hypotenuse is always equal to the Square root of the addition of square of opposite side and adjacent side in any right triangle.
r2 = a2 + b2
Just like this opposite and adjacent also can be find by this formula:
a2 = r2 - b2
And
b2= r2 - a2
These are some Trigonometric Functions of acute angles.