Trigonometry is all about angles and their measurement. When discussing the various trigonometric functions, we keep in mind the formula of compound angles to give accurate results. The chapter below discusses the same idea of measurement in Trigonometry.
Trigonometric Functions for Compound Angles
Trigonometry is the branch of geometry that deals with measuring the sides of a triangle. We developed trigonometry to solve problems and measurements involving triangles. Navigators, Engineers, Scientists, Seismologist and Meteorologists use Trigonometry in various applications.
The latest use of trigonometry is in the study of atoms which in the long run is going to prove a masterstroke of humanity. The subject so wide yet compact has varied ratios and functions that help in the better learning of angles. Before delving deeper into the subject let’s discuss the concept of angles and compound angles:
Browse more Topics under Trigonometric Functions
- Measurement of Angles
- Introduction to Trigonometric Functions
- Domain and Range of Trigonometric functions
- Trigonometric Equations
When a ray rotates about its initial point, it forms an angle. The original point from where the rotation initiates is called the initial side, while the position after rotation, where the ray stops is called the terminal side. The central point of rotation is called the vertex. If a ray rotates in an anticlockwise direction, it forms a positive angle.
If the ray rotates in clockwise direction, it forms a negative angle.
An angle can be measured from the difference of positions after rotation. Angle thus is the measurement of a single rotation from the initial side to the terminal side and can be measured in degree or radians.
The Relation between Degree and Radians
Units used for measuring Angles are Degrees or Radians. When we denote an angle in θ° we are measuring the angle in terms of degree, while when we denote it with β we are measuring it in terms of a radian. We generally omit the use of the word Radian when we measure it in radians.
- Radian measure= π /180 × Degree measure
- Degree measure= 180/π × Radian measure
A compound angle is an algebraic sum of two or more angles. We use trigonometric identities to connote compound angles through trigonometric functions. The sum and difference of functions in trigonometry can be solved using the compound angle formula or the addition formula. Here, we shall deal with functions like (A+B) and (A-B). The formula for trigonometric ratios of compound angles are as follows:
- sin (A + B) = sin A cos B + cos A sin B
- sin (A – B) = sinA cosB – cosA sinB
- cos (A + B) = cosA cosB – sinA cosB
- cos (A – B) = cosA cosB + sinA cosB
- tan (A + B) = [tanA + tanB] / [1 – tanA tanB]
- 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.
Solved Examples for You
Question 1: In an acute-angled triangle, cot B . cot C + cot A . cot C + cot A . cot B =
Answer : cotB . cotC + cotA . cotC + cotA . cotB
⇒cot(A+B) = cot(π−C)
⇒cotA cotB − 1cotB + cotA = − cotC
⇒cotA . cotB + cotB . cotC + cotA . cotC = 1
Question 2: What are compound angle formulas?
Answer: A compound angle formula or addition formula is basically a trigonometric identity that expresses a trigonometric function of (A+B) or (A−B) in expressions of trigonometric functions of A and B.
Question 3: What is a bevel cut?
Answer: A bevel cut is a cut you see at an angle other than 90 degrees along with the thickness of the material. In other words, it is a cut which is angled relative to the face of a material. Moreover, we frequently measure the angle against a square-edge cut.
Question 4: What is a compound cut?
Answer: A compound cut comprises of two angles. They are the bevel angle and the miter angle. The bevel angle (or blade tilt) is basically the tilt of the saw blade from vertical on the saw table. Similarly, the miter angle is set on the miter gauge of the table saw. Moreover, a perpendicular cut is having a mitre of 0°.
Question 5: What is trigonometry?
Answer: Trigonometry refers to the branch of geometry that is dealing with measuring the sides of a triangle. Trigonometry was developed for solving problems and measurements that involve triangles. Moreover, navigators, scientists, meteorologists, engineers and more use it for various purposes.