Formula, Proof, Examples

     

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 lớn most of the appliances which come in daily use. It is used for programming, computing, navigating, medical imaging, measuring the heights of buildings and mountains, etc.

Bạn đang xem: Formula, proof, examples

Trigonometry is a branch of standardized mathematics that đơn hàng with the relationship between lengths, heights, và angles.

Trigonometry includes its own trigonometric functions, expressions, angles, và their values which are used to 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 to lớn 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, và 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:

ExpressionDerived Formula
sin (A + B)sinAcosB + cosAsinB
sin(A – B)sinAcosB – cosAsinB
cos(A + B)cosAcosB – sinAsinB
cos(A – B)cosAcosB + sinAsinB
tan(A + B)fractanA+tanB1-tanAtanB
tan(A – B)fractanA-tanB1+tanAtanB

What is the formula of tan(A – B)?

Tan(A – B) is the sixth formula among the six addition formulae of trigonometry used lớn conduct trigonometric calculations. The formula is derived with the help of the previous four additional formulae of sine & cosine.

Xem thêm: Em Hãy Kể Về Một Chuyến Về Quê Hay Nhất ❤️️15 Bài Văn Điểm 10

The result for tan(A – B) is derived in the terms of sines và cosines. The trigonometric formula is given by:

tan(A-B)=fractanA-tanB1+tanAtanB

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

tan(A-B)=fracsin(A-B)cos(A-B)

By putting in the formulae of sin(A-B) và cos(A-B), the result is

=fracsinAcosB-cosAsinBcosAcosB-SinAsinB

Now, dividing every term of right hand side with cosAcosB we get,

=fracfracsinAcosBcosAcosB-fraccosAsinBcosAcosBfraccosAcosBcosAcosB+fracsinAsinBcosAcosB

Canceling out the common factors we get,

=fracfracsinAcosA-fracsinBcosB1+fracsinAsinBcosAcosB

As we have rã = sin/cos

=fractanA-tanB1+tanAtanB

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.

Sample Questions

Question 1: Find an expression fortan 15°?

Answer:

From the question we can assume the expression 15° = 60° – 45°.

Xem thêm: Ảnh Bác Hồ Quàng Khăn Đỏ Cho Thiếu Nhi, Về Bức Ảnh Quý Bác Hồ Với Đội Viên

The value of tan60°=√3 andtan45° = 1

Now, following the expression

tan15° = tan(60° – 45°)

=frac

=fracsqrt3-1sqrt3+1

Multiplying both numerator & denominator by√3-1

fracsqrt3-1sqrt3+1=fracsqrt3-1sqrt3+1×fracsqrt3-1sqrt3-1

=frac3-sqrt3-sqrt3+13-1

= 2 – √3

Question 2: If the tanA = 1/2 and tanB = 1/3, find the value for expression tan(A + B).