prepare formula chart on trigonometry
Answers
In a right angled triangle, we have basically 3 sides namely – Hypotenuse, Opposite side and Adjacent side. The longest side is known as the hypotenuse, the side opposite to hypotenuse is opposite and the side where both hypotenuse and opposite rests is the adjacent side.
There are basically 6 Laws used for finding the elements in Trigonometry. They are called trigonometric functions.
The six trigonometric functions are sine, cosine, secant, co-secant, tangent and co-tangent. By using the above right angled triangle as reference, the trigonometric functions or trigonometric identities are derived:
Sine=\frac{Opposite}{Hypotenuse}
Secant=\frac{Hypotenuse}{Adjacent}
Cosine=\frac{Adjacent}{Hypotenuse}
Tangent=\frac{Opposite}{Adjacent}
Co−Secant=\frac{Hypotenuse}{Opposite}
Co−Tangent=\frac{Adjacent}{Opposite}
The reciprocal identities are given as:
Cosec\Theta =\frac{1}{sin\Theta }
sec\Theta =\frac{1}{cos\Theta }
cot\Theta =\frac{1}{tan\Theta }
sin\Theta =\frac{1}{Cosec\Theta }
cos\Theta =\frac{1}{sec\Theta }
tan\Theta =\frac{1}{cot\Theta }
All these are taken from a right angled triangle. With the length and base side of the right triangle given, we can find out the sine, cosine, tangent, secant, cosecant and cotangent values using trigonometric formulas. The reciprocal trigonometric identities are also derived by using the trigonometric functions.
Basic Trigonometry Formulas:
A.Trigonometry Formulas involving Periodicity Identities:
sin(x+2\pi )=sin\; xcos(x+2\pi )=cos\; xtan(x+\pi )=tan\; xcot(x+\pi )=cot\; x
All trigonometric identities are cyclic in nature. They repeat themselves after this periodicity constant. This periodicity constant is different for different trigonometric identity.
tan 45 = tan 225 but this is true for cos 45 and cos 225.
Refer to the abovetrigonometry table to verify the values.
B.Trigonometry Formulas involving Cofunction Identities – degree:
sin(90^{\circ}-x)=cos\; xcos(90^{\circ}-x)=sin\; xtan(90^{\circ}-x)=cot\; xcot(90^{\circ}-x)=tan\; x
C.Trigonometry Formulas involving Sum/Difference Identities:
sin (x + y) = sin(x) cos(y) + cos(x) sin(y)cos(x + y) = cos(x) cos(y) – sin(x) sin(y)tan(x+y)=\frac{tan\: x+tan\: y}{1-tan\: x\cdot tan\: y}sin(x – y) = sin(x) cos(y) – cos(x) sin(y)cos(x – y) = cos(x) cos(y) + sin(x) sin(y)tan(x-y)=\frac{tan\: x-tan\: y}{1+tan\: x\cdot tan\: y}
D.Trigonometry Formulas involving Double Angle Identities:
sin(2x) = 2 sin(x).cos(x)cos(2x) = cos^{2}(x) – sin^{2}(x),cos(2x) = 2 cos^{2}(x) -1cos(2x) = 1 – 2 sin^{2}(x)tan(2x) = \frac{[2\: tan(x)]}{[1-tan^{2}(x)]}
E.Trigonometry Formulas involving Half Angle Identities:
sin\frac{x}{2}=\pm \sqrt{\frac{1-cos\: x}{2}}cos\frac{x}{2}=\pm \sqrt{\frac{1+cos\: x}{2}}tan(\frac{x}{2}) = \sqrt{\frac{1-cos(x)}{1+cos(x)}}
Also, tan(\frac{x}{2}) = \sqrt{\frac{1-cos(x)}{1+cos(x)}}\\ \\ \\ =\sqrt{\frac{(1-cos(x))(1-cos(x))}{(1+cos(x))(1-cos(x))}}\\ \\ \\ =\sqrt{\frac{(1-cos(x))^{2}}{1-cos^{2}(x)}}\\ \\ \\ =\sqrt{\frac{(1-cos(x))^{2}}{sin^{2}(x)}}\\ \\ \\ =\frac{1-cos(x)}{sin(x)}
So, tan(\frac{x}{2}) =\frac{1-cos(x)}{sin(x)}
F.Trigonometry Formulas involving Product identities:
sin\: x\cdot cos\:y=\frac{sin(x+y)+sin(x-y)}{2}cos\: x\cdot cos\:y=\frac{cos(x+y)+cos(x-y)}{2}sin\: x\cdot sin\:y=\frac{cos(x+y)-cos(x-y)}{2}
G.Trigonometry Formulas involving Sum to Product Identities:
sin\: x+sin\: y=2sin\frac{x+y}{2}cos\frac{x-y}{2}sin\: x-sin\: y=2cos\frac{x+y}{2}sin\frac{x-y}{2}cos\: x+cos\: y=2cos\frac{x+y}{2}cos\frac{x-y}{2}cos\: x-cos\: y=-2sin\frac{x+y}{2}sin\frac{x-y}{2}
Answer:
There are many different kind of identities in Trigo.
A}
1) sinθ÷cosθ = tanθ
2) sin²θ + cos²θ = 1
3) 1 + tan²θ = sec²θ
4) 1 = cot²θ = cosec²θ
B}
1) sin ( 90 - θ ) = cos∅
2) cos ( 90 - θ ) = sin
3) sin ( 180 - θ ) = sinθ
4) cos ( 180 - θ ) = cosθ
C}
1) sin²θ = 1 - cos2θ ÷ 2
2) cos²θ = 1 + cos2θ ÷ 2
tan²θ = 1 - cos2θ ÷ 1 + cos2θ
Additional Information:
Trigonometric Identities of specific angles.
θ 0° 30° 45° 60° 90°
sin 0 1/2 1 /√2 √3/2 1
cos 1 √3/2 1/√2 1/2 0
tan 0 1/√3 1 √3 ∞
cot ∞ √3 1 1/√3 0
sec 1 2/√3 √2 2 ∞
cosec ∞ 2 √2 2/√3 1