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What is trigonometry?
Trigonometry is the branch of mathematics to study the relationship between the sides and angles of a right-angled triangle. Trigonometry comes from the Greek words ‘trigon’ (meaning, triangle) and ‘metron’ (meaning, measure).
In a right-angled triangle, the three sides are perpendicular, the base (adjacent) and the hypotenuse (opposite)
In a right-angled triangle ABC:
The line joining the two angles is called the base.
The front side of the acute angle is called perpendicular.
The front side of the right angle is called the hypotenuse.
There are six functions of the angle used in trigonometry. They are: sine (sin), cosine (cos), tangent (tan), cosecant (cosec), secant (sec) and cotangent (cot).
Sinθ = Perpendicular/Hypotenuse
Cosθ = Base/Hypotenuse
Tanθ = (Perpendicular )/Base
Cosecθ = Hypotenuse/Perpendicular
Secθ = Hypotenuse/Base
Cotθ = Base/Perpendicular
In mathematics, the trigonometric function is also known as the circular function.
Why are trigonometric functions sometimes called circular functions?
Trigonometric functions are called circular functions because function values can be expressed in terms of coordinates of a point on a unit circle.
These trigonometric functions are called circular functions because radian measures of angles are determined by the lengths of arcs of circles.
x2+y2=1 is the equation of a circle.
The graph of the equation x2+y2=1 is a circle in the rectangular coordinate system. This graph is called a unit circle with the centre (0, 0) and a radius of 1 unit.
θ measured in radian.
Let q be any real number.
The value of q is negative in the clockwise direction and positive in the anticlockwise direction
Start from A and measure the q unit in an anticlockwise direction.
Sin q = y
Cos q = x
tan q=sin qcos q; q ≠ 0.
The sign of all six trigonometric ratios depends on the length of arc q.
Sign of trigonometry function in all four quadrants:
Periodic function:
The circumference of the unit circle is 2πr.
If a point P travels around a unit circle, after each 2π distance, it repeats itself.
In other words, the values of coordination of P remain unchanged if 2π is added or subtracted.
For example:
Sin (q+2π) = sin q
Sin (q-2π) = sin q
Cos (q-2π) = cos q
Cos (q+2π) = cos q
This property of functions is called periodic function.
Example:
The value of sin 30°=1/2 , then sin 390°=?
Solution: sin 390° is written as sin (30°+360°)
sin (30°+360°) = sin(30°+2π) = sin 30°=1/2
Even and odd trigonometric function
f is even function if f(-x) = f(x)
f is odd function if f(-x) = -f(x)
The graph of the even function will be symmetric about the y-axis.
The graph of the odd function will be symmetric about the x-axis.
Graphs
Sine function:
Degrees 0° 30° 45° 60° 90°
Radians 0 π/6 π/4 π/3 π/2
sin x 0 0.500 0.707 0.866 1
The period of sine function is 2π.
Cosine function:
Degrees 0° 30° 45° 60° 90°
Radians 0 π/6 π/4 π/3 π/2
cos x 0 0.866 0.707 0.500 1
The period of cosine function is 2π.
Tangent function:
The tangent is odd function because
tan(-x)=(sin(-x))/(cos(-x))=(-sin x)/cos x=-tan x
The period of tangent function is π
Cosecant function:
The period of cosecant function is 2π.
Conclusion
Trigonometry can be used in many areas such as astronomy and architecture and can aid in calculating many things. It’s a useful supplement to geometry and actual measurements. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures. Trigonometry developed from a need to compute angles and distances in fields as diverse as astronomy, surveying, artillery range finding and mapmaking.
