Trigonometry has always been part of advanced mathematics which is applicable in almost all fields whether it be architecture, geography, physics, astronomy, investigation, or daily life application. Although trigonometry is not used directly in real life it is applied to most of the appliances which come in daily use. It is used for programming, computing, navigating, medical imaging, measuring the heights of buildings & mountains, etc.
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Trigonometry is a branch of standardized mathematics that đơn hàng with the relationship between lengths, heights, & angles.
Trigonometry includes its own trigonometric functions, expressions, angles, và their values which are used lớn solve trigonometric problems.
Addition Formulas of Trigonometry
Generally, there are six addition formulae in trigonometry. These all six formulae are interconnected as one is used khổng lồ derive the other. These formulae are applicable for solving trigonometric problems. The first two addition formulae are of sine, the second two are related to cosine, and the third pair is of the tangent which is derived from four previous formulae.
Look into all the six formulae in the below given table:
|sin (A + B)||sinAcosB + cosAsinB|
|sin(A – B)||sinAcosB – cosAsinB|
|cos(A + B)||cosAcosB – sinAsinB|
|cos(A – B)||cosAcosB + sinAsinB|
|tan(A + B)|
|tan(A – B)|
What is the formula of tan(A – B)?
Tan(A – B) is the sixth formula among the six addition formulae of trigonometry used to conduct trigonometric calculations. The formula is derived with the help of the previous four additional formulae of sine & cosine.
The result for tan(A – B) is derived in the terms of sines and cosines. The trigonometric formula is given by:
Now, let’s take a look into how the formula is derived from previous addition formulas of sines & cosines.
Derivation of the Formula
The formula can be derived from previous addition formulae as we know that
By putting in the formulae of sin(A-B) và cos(A-B), the result is
Now, dividing every term of right hand side with cosAcosB we get,
Canceling out the common factors we get,
As we have tan = sin/cos
The same method of derivation is used to lớn derive tan(A + B) from the other formulae.
Here, are some trigonometric problems using the tan(A – B) formula.
Question 1: Find an expression fortan 15°?
From the question we can assume the expression 15° = 60° – 45°.
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The value of tan60°=√3 andtan45° = 1
Now, following the expression
tan15° = tan(60° – 45°)
Multiplying both numerator và denominator by√3-1
= 2 – √3
Question 2: If the tanA = 1/2 and tanB = 1/3, find the value for expression tan(A + B).