Circular Functions in Trig

In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle. The most familiar trigonometric functions are the sine, cosine, and tangent. The sine function takes an angle and tells the length of the y-component of that triangle (Source: Wikipedia)

Circular function in trig

Right triangle for all time contain a 90° angle, now denoted at C. Angles A and B might differ. Trigonometric functions identify the relations between side lengths also interior angles of a right triangle.
Illustrate the trigonometric functions used for the angle A.
Establish through either right triangle to include the angle A.
Three sides of the triangle are,
Adjacent side:
It is the side specifically in contact through both the angle we are concerned within (angle A).
Opposite side:
It is the side opposite toward the angle we are concerned within (angle A)
Hypotenuse side:
It is the side opposed the right angle. The hypotenuse is the highest part of a right-angled triangle.



Trigonometric functions:
 Sin A = opposite / hypotenuse.
Cos A = adjacent/ hypotenuse
Tan A = opposite /adjacent
Csc A =1/Sin A= hypotenuse/ opposite
Sec A =1/Cos A= hypotenuse/ adjacent
Cot A =1/Tan A adjacent / opposite
The circular function has the following equations
Let u = x + y and v = x - y. Then the three equations yields the sums otherwise differences
sin(u) + sin(v) = 2 sin(`(u + v) / 2` ) cos(`(u - v) / 2` )
cos(u) + cos(v) = 2 cos(`(u + v) / 2` ) cos(`(u - v) / 2` )
cos(v) - cos(u) = 2 sin(`(u + v) / 2` ) sin(`(u - v) / 2` )

Examples for circular function in trig

Example 1 for circular function:
Apply the tangent ratio toward solve the unidentified side of the triangle?
Given angle of triangle are 40⁰ and opposite side of triangle are 16.
Solution:
Specified to angle of triangle is 40⁰ and opposite side of triangle is 16. Find out the adjacent side
 tan 40⁰ = opposite/adjacent
tan 40⁰ x = 16
x = 16/tan 40⁰
x =16/0.8391     {since the value of tan 40 degree is 0.8391}
x=19.06
The value of adjacent side =19.06
Example 2 for circular function:
If u=30 and v=30 then find out the sin (u) + sin (v)?
Solution:
We know the formula,
sin (u) + sin(v) = 2 sin(`(u + v) / 2` ) cos (`(u - v) / 2` )
                         =2 sin(`(30+30)/2` ) cos(30-30)
                         =2 sin(60/2) cos 0
                         =2 sin30 cos 0
                         =2x(1/2)x1
 sin (u) + sin(v) =1