Half Angle Formulas
Half Angle formulas are developed for the trigonometric functions sin, cos and tan. Half angle formulas are very much useful in solving trigonometric equations. For example, m sin x + n cos x = p, where m, n, p are constants. These formulas are very helpful in integrating rational functions.
Let us see how to develop Trig Half Angle Formula for tan and cosine.
Tan Half Angle Formula
The concept for developing the tan half angle formula starts with dividing tan by half, say tan M/2. The double angle formula and various trigonometric ratios are used to develop the half angle formula.
Tan M/2 is given by (Sin M/2) /(Cos M/2), which is given by finding the square root of the value obtained by dividing 1-Cos M by 1+Cos M. Only in case of tan, the next step is to avoid the square root. This can be done in two methods.
Method 1:
Multiply the denominator and numerator by square root of 1+Cos M and then apply the formula sin^2 x + cos^2 x equal to one. We get the equation for half angle. The half angle Tan M/2 is given by Sin M divided by 1 + Cos M.
Method 2:
Instead of square root of (1+Cos M), multiply the denominator and numerator by square root of 1-Cos M and then apply the formula sin^2 x + cos^2 x equal to one. We get the equation for half angle. In this method, the half angle Tan M/2 is given by 1 - Cos M divided by Sin M.
From these two methods, the trigonometric half angle formula for tan is given by:
Tan (M/2) = Sin M / (1+Cos M) = (1 - Cos M)/Sin M.
An important point to be noted is that Cos M always lies between -1 and +1. Therefore both 1+cos M and 1-cos M is positive. The sign of the trigonometric function Sine of an angle is the same as that of the tan of half angle.
Cosine Half Angle Formula
The first step to obtain the cosine half angle formula starts from the formula of cosine of a double angle. Cos 2M is given by one subtracted from 2cos^2M. By substituting n/2 in place of M where n = 2M and then exchanging the values of left to the right and right to the left followed by an addition of 1 on both sides and dividing the result on both sides by 2, we get the value of Cos n/2 as plus or minus square root of the value obtained by dividing (1+cos n) by 2. The sign of cosine half angle i.e. cos n/2 depends upon the quadrant where it is located.
In first and fourth quadrant, the formula is positive i.e. cos n/2 = sqrt ((1+cos n)/2).
In second and third quadrant, the formula is negative i.e. cos n/2 = - sqrt ((1+cos n)/2.
This gives the trigonometric half angle formula for cosine.
Learn more on about What is Half Angle Formula? and its Examples.Between, if you have problem in learning and solving trigonometry find trigonometry problem solving help related resources.
Half Angle formulas are developed for the trigonometric functions sin, cos and tan. Half angle formulas are very much useful in solving trigonometric equations. For example, m sin x + n cos x = p, where m, n, p are constants. These formulas are very helpful in integrating rational functions.
Let us see how to develop Trig Half Angle Formula for tan and cosine.
Tan Half Angle Formula
The concept for developing the tan half angle formula starts with dividing tan by half, say tan M/2. The double angle formula and various trigonometric ratios are used to develop the half angle formula.
Tan M/2 is given by (Sin M/2) /(Cos M/2), which is given by finding the square root of the value obtained by dividing 1-Cos M by 1+Cos M. Only in case of tan, the next step is to avoid the square root. This can be done in two methods.
Method 1:
Multiply the denominator and numerator by square root of 1+Cos M and then apply the formula sin^2 x + cos^2 x equal to one. We get the equation for half angle. The half angle Tan M/2 is given by Sin M divided by 1 + Cos M.
Method 2:
Instead of square root of (1+Cos M), multiply the denominator and numerator by square root of 1-Cos M and then apply the formula sin^2 x + cos^2 x equal to one. We get the equation for half angle. In this method, the half angle Tan M/2 is given by 1 - Cos M divided by Sin M.
From these two methods, the trigonometric half angle formula for tan is given by:
Tan (M/2) = Sin M / (1+Cos M) = (1 - Cos M)/Sin M.
An important point to be noted is that Cos M always lies between -1 and +1. Therefore both 1+cos M and 1-cos M is positive. The sign of the trigonometric function Sine of an angle is the same as that of the tan of half angle.
Cosine Half Angle Formula
The first step to obtain the cosine half angle formula starts from the formula of cosine of a double angle. Cos 2M is given by one subtracted from 2cos^2M. By substituting n/2 in place of M where n = 2M and then exchanging the values of left to the right and right to the left followed by an addition of 1 on both sides and dividing the result on both sides by 2, we get the value of Cos n/2 as plus or minus square root of the value obtained by dividing (1+cos n) by 2. The sign of cosine half angle i.e. cos n/2 depends upon the quadrant where it is located.
In first and fourth quadrant, the formula is positive i.e. cos n/2 = sqrt ((1+cos n)/2).
In second and third quadrant, the formula is negative i.e. cos n/2 = - sqrt ((1+cos n)/2.
This gives the trigonometric half angle formula for cosine.
Learn more on about What is Half Angle Formula? and its Examples.Between, if you have problem in learning and solving trigonometry find trigonometry problem solving help related resources.
