Introduction to preparation for trigonometry example

In the preparation of trigonometry example we need to learn three basic trigonometric functions (trigonometric ratios) are: Tangent, Sine and Cosine.

Trigonometry preparation example deals with the angle and side of triangles. For preparation unknown angles or lengths are calculated by using trigonometry ratios such as sine, cosine,
and tangent.The word ‘trigonometry’ stands for ‘triangle measurement’.

Preparation in trigonometry the angles of a triangle are important. For this reason, trigonometry example is very closely linked with geometry. While preparing trigonometric example
functions are fundamentally related, they can be used to find out the dimensions of any triangle given limited information.
preparation of trignometric examples
Important relations for tangent, cotangent, secant, and cosecant in terms of sine and cosine.
tan t = `sin t /cos t`
Sec t = `1/cos t`
Cot t = `1/ tan t` = `cos t/sin t`
Cosec t = `1/sin t`
The Pythagorean formula for sines and cosines.
Sin^{2} t + cos^{2} t = 1
Identity expressing trigonometric functions in terms of their complements
Cos t = sin (`1/2` – t) sin t = cos (`1/2` – t)
Cot t = tan ( `1/2` – t) tan t = cot(`1/2` – t)
Cosec t = sec (`1/2` – t) sec t = cosec (`1/2` – t)
Examples problems on preparation for trign
Q:1 Find the principle value of sin^{1}(`1/sqrt2` )
Sol:
Let sin^{1}(`1/sqrt 2` )=y.Then sin y=`1/sqrt 2`
We know that the range of the principal value branch of sin^{1} is(`pi/2` ,`pi/2` )and
Sin (`pi/4` ) =`1/sqrt2` .Therefore, principal value of sin^{1}(`1/sqrt2` ) is `pi/4`
Q:2 Prove that
`(Cos (t) +sin (t))/(Cos (t)sin (t)) ` = ` ( cos (2t))/ (1sin (2t))`
solution:
` (Cos (2t) )/(1 sin (2t)) ` = ` (cos2(t)sin2(t))/(12sin(t)cos(t))` = `
((cos(t)sin(t))(cos(t)+sin(t)))/ (cos(t)cos(t)2sin(t)cos(t)+sin(t)sin(t))`
= ` ((Cos (t)sin (t))(Cos (t) +sin (t)))/((Cos (t)sin (t))(Cos (t)sin (t)))`
`(Cos (t) + sin (t))/(Cos (t)  sin (t))`
Q:3 Which quadrant is the fatal side of an angle of 750 degrees located?
Sol:
750 degrees = 360 degrees + 360 degrees + 30 degrees.
Hence an angle of 750 degrees, in normal position, has its terminal side in quadrant one.