Angle Formula
Every time when two rays intersect or half-lines projecting the common endpoints then the corner points of angles are named as vertices or angles of the rays are named as sides. Angle is also termed as the measurement of a turn between any two lines. The unit of an angle is degree or radian. Further, angles could be divided into multiple categories like double-angle formula, half angle formula, compound angle, or interior angle etc. The angle formula in mathematics is given as below –
Double Angle Formulas
When multiple angles are expanded then it will make double angles and take the sum of different angles then again apply the double angle formula.
\[\ sin(A+B)=sinA\;cosB+cosA\;sinB\]
\[\ sin(A-B)=sinA\;cosB-cosA\;sinB\]
\[\ cos(A+B)=cosA\;cosB-sinA\;sinB\]
\[\ cos(A-B)=cosA\;cosB+sinA\;sinB\]
\[\ sin\alpha +sin\beta =2sin\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}\]
\[\ sin\alpha -sin\beta =2sin\frac{\alpha -\beta }{2}cos\frac{\alpha +\beta }{2}\]
\[\ cos\alpha +cos\beta =2cos\frac{\alpha +\beta }{2}cos\frac{\alpha -\beta }{2}\]
\[\ cos\alpha -cos\beta =-2sin\frac{\alpha +\beta }{2}sin\frac{\alpha -\beta }{2}\]
\[\ sin2\alpha =2\;sin\alpha\;cos\alpha\]
\[\ cos2\alpha =cos^{2}\alpha -sin^{2}\alpha = 2cos^{2}\alpha -1=1-2sin^{2}\alpha\]
\[\ tan2\alpha =\frac{2tan\alpha }{1-tan^{2}\alpha }\]
Half Angle formula
In case of special identities where sum and differences of sine and cosine functions are calculated, it would be termed as double angle identities or half angle identities. Any double angle when divided by two, the half-angle formula can be derived as given below.
\[\ Sine\;of\;a\;Half\;Angle = \sin \frac{a}{2} = \pm \sqrt{\frac{(1- \cos a)}{2}}\]
\[\ Cosine\;of\;a\;Half\;Angle = \cos \frac{a}{2} = \pm \sqrt{\frac{(1+ \cos a)}{2}}\]
\[\ Tangent\;of\;a\;Half\;Angle = \tan \left ( \frac{a}{2} \right ) = \frac{1 – \cos a}{\sin a} = \frac{\sin a}{1 + \cos a}\]
First Trigonometric expression is an example of double angle formula and the second equation is an example of half-angle formula. In the same way, their multiple halves – angle formulas can be derived for multiple trigonometric functions one by one.
Multiple Angle Formulas
The trigonometric functions for multiple angles are named as multiple angle formula. For double or triple angles formulas, there would come multiple angle formulas ahead. The popular Trigonometric functions are Sine, Cosine, Tangent etc.
The sin formula for multiple angle is:
\[\large sin \theta = \sum_{k=0}^{n}\;cos^{k}\theta \; Sin^{n-k}\theta\; Sin\left [\frac{1}{2}\left(n-k\right)\right]\pi\]
Where n=1,2,3,……
General formulas are,
\[\large sin^{2}\theta =2 \times cos\,\theta \; sin\,\theta\]
\[\large sin^{3}\theta =3 \times cos^{2}\,\theta \; sin\, \theta \; sin^{3}\,\theta\]
The multiple angle’s Cosine formula is given below:
\[\large Cos\;n\, \theta =\sum_{k=0}^{n}cos^{k}\theta \,sin^{n-k}\theta \;cos\left [\frac{1}{2}\left(n-k\right)\pi\right]\]
Where n = 1,2,3
The general formula goes as:
\[\large cos^{2}\, \theta =cos^{2}\, \theta – sin^{2}\, \theta\]
\[\large cos^{3}\, \theta =cos^{3}\, \theta – cos\, \theta \; sin^{2}\, \theta\]
Tangent Multiple Angles formula
\[\large Tan\;n\theta = \frac{sin\;n\theta}{cos\;n\theta}\]
Right Angle Formula
The three sides for a right-angle triangle in mathematics are given as Perpendicular, Base, and the Hypotenuse. The largest side that is opposite to the right angle will be termed as the Hypotenuse. To find a particular side of a Triangle, we should know the other two sides of the Triangle. And the formula is given as –
\[\large Hypotenuse^{2}=(Adjacent\;Side)^{2}+(Opposite\;Side)^{2}\]
The other popular name for right angle formula is the Pythagorean theorem and a right angle is an angle that exactly measures 90-degree. This is the most used formula is mathematics and should be clearly understood by students when preparing for higher studies or competitive exams. There is a special notation in mathematics for the right-angle and it is given by a small square between two sides. Let us understand through figure how it looks alike –
The right angles could be seen at multiple places in our daily life. For example, every rectangular or square object you see around you is a right angle. One of the most common places forthe right angle is a triangle. If there are no right-angles, then Trigonometry existence is not possible in this case. All Trigonometry concepts are based on the right-angle formulas only. Also, the right-angle formula has multiple applications in real-life too.
For example, when you want to calculate the distance up to the slope or you wanted to measure the height of a hill, only right-angle triangle formulas are useful. In the same way, there are just the endless applications for right-angle formula in mathematics.
Side Angle Side Formula
There are three popular steps for side angle side formulas. These are –
- First, you should use the low of Cosine to calculate the unknown side.
- In the second step, you should find the smallest of two angles.
- Now add the three angles to 180-degrees and calculate the third one.
\[\ Area=\frac{ab\;Sin\,C}{2}\]