Trigonometric Ratios of Compound Angles

Trigonometry is the area of mathematics that examines angles and the various ways in which they can be measured. The usage of trigonometric functions, such as trigonometric ratios, compound angles, and multiple angles, is necessary in order to obtain conclusions that are consistent. This page is a discussion on the trigonometric ratios of compound angles, complete with cases that have been solved.

A ray that is rotated about an initial point is an example of an angle. The side that faces away from where the rotation begins is known as the initial side, while the side that faces toward where it ends is known as the terminal side. When a ray is turned in the anticlockwise direction, the resulting angle is positive, whereas the angle produced by rotating the ray in the clockwise direction is negative. Both degrees and radians are valid units for measuring angles.

Compound Angles

A compound angle is a name given to the angle that is the algebraic sum of two or more angles. Compound angles can be represented by their corresponding trigonometric identities. Using the idea of compound angles, it is possible to compute the fundamental arithmetic operations of calculating the sum and difference of functions.

List of Trigonometric Ratios of Compound Angles

The following are the formulas for calculating the trigonometric ratios of compound angles:

• sin (A + B) = sin A cos B + cos A sin B
• sin (A – B) = sin A cos B – cos A sin B
• cos (A + B) = cos A cos B – sin A cos B
• cos (A – B) = cos A cos B + sin A cos B
• tan (A + B) = [tan A + tanB] / [1 – tan A tan B]
• tan (A – B) = [tan A – tan B] / [1 + tan A tan B]
• sin (A + B) sin (A – B) = sin2 A – sin2 B = cos2 B – cos2 A
• cos (A + B) cos (A – B) = cos2 A – sin2 A – sin2 B = cos2 B – sin2 A

Transformation of Products into Sum or Difference of Sines and Cosines:

•  2 sin A cos B = sin (A+B) + sin (A-B)
•  2 cos A cos B = cos (A+B) + cos (A-B)
•  2 cos A sin B = sin (A+B) – sin (A-B)
•  2 sin A sin B = cos (A-B) – cos (A+B)

Multiple and submultiple Angles

(a) sin 2A = 2 sin A cos A = 2 tan A/(1+tan2A)

(b) cos 2A = cos2A – sin2A  = 2cos2A-1 = 1-2 sin2A

(c) tan 2A = 2 tan A/(1-tan2A)

(d) sin 3A = 3 sinA – 4 sin3A

(e) cos 3A = 4 cos3A – 3 cos A

(f) tan 3A = (3 tan A- tan3A)/(1-3tan2A)

Proofs for sin(A+B), cos(A+B), and tan(A+B)

The process of finding a solution will be aided by the following figure.

Throughout the course of the explanation, we will be utilising and correspondingly in place of ‘A and ‘B wherever appropriate.

The following diagram depicts five right triangles: △APB, △PFC, △AFD, △DEF and △AEF . Within each of these triangles, EF||PBC, EF=BC, and CF=BE.

AS EF||PB, ∠CPF=∠EFD=α. Once more, in the right EFD, ∠α+∠x=900, and in the right AFD, ∠x+∠DAF=900, ∠DAF=∠α.

With this background, we may deduce sin(+).

Proof of sin(A+B)

sin(α+θ)=ABAP

=AE+BEAP

=CFAP+AEAP

=CFPF.PFAP+AEAF.AFAP

=sinα.cosθ+cosα.sinθ.

Proof of cos(A+B)

Using the same figure, we have,

cos(α+θ)=PBAP

=PC−BCAP

=PCAP−EFAP

=PCPF.PFAP−EFAF.AFAP

=cosα.cosθ−sinα.sinθ.

Proof of tan(A+B)

We’ll prove tan(+α+θ) using sin(+α+θ) and cos(+α+θ) instead of deriving it.

tan(α+θ)=tanα+tanθ/1−tanα.tanθ.

tan(α+θ)=sin(α+θ)/cos(α+θ)

=sinα.cosθ+cosα.sinθ/cosα.cosθ−sinα.sinθ.

After removing the factorcosα.cosθ from both the numerator and the denominator of the right-hand side of the preceding relation, we have the following:

tan(α+θ)=tanα+tanθ/1−tanα.tanθ.

The derivation of cos(A+B) follows the same pattern as the derivation of sin(A+B), and the expression for tan(A+B) may be simply derived from sin(A+B) and cos(A+B). Therefore, it is important for us to grasp the derivation of sin(A+B).

Conclusion

Trigonometry is an extremely helpful tool for determining which side of a particular triangle is unknown. The idea of the right triangle, which is employed in trigonometry, is put to excellent use in this particular fieldwork exercise. We are able to become proficient in reading the vertical angle using the theodolite, which enables us to accurately determine the height of the Mapuaadmin building.

Students who have a solid grasp of trigonometry ratios are able to calculate out complicated angles and dimensions in a comparatively short amount of time. Trigonometry is one of the most useful subfields of mathematics due to its widespread application in the fields of engineering, architecture, and a variety of other scientific disciplines.