Trigonometry: Academic Content Test

                                   TRIGONOMETRY                   

Trigonometry was invented primarily for astronomy, that is, for spherical geometry. The Babylonians were the first to give coordinates for stars. They used the ecliptic as their base circle in the celestial sphere, that is, the crystal sphere of stars. Determining those coordinates of stars (and planets) is not so easy. You can measure the elevation of a star directly, but that’s constantly changing. You can also measure the angle between two stars at a time. Trigonometry can be used to determine those coordinates.

INTRODUCTION

Trigonometry is a long and off-putting name for what is really a fun subject. A Trigon is a fancy name for a triangle; analogous to the words octagon or pentagon, Metry refers to measurement. So trigonometry means either measuring triangles, or using triangles to measure other things.

BASIC TRIGONOMETRIC RESULTS

One of the mysteries of trigonometry is: Why does every one of the six ratios of side lengths in a right triangle have its own special name? Why for example does 1/sin x ​ have a name of its own? 

                                      

Suppose our angle theta, as in the picture here lies between the x axis and the line 0B. The ancients drew a line segment that extends from the point B tangent to the unit circle to the x axis at point C. The length of this segment they called the tangent of the angle θ. (When the line has a positive slope the tangent is taken to be negative.) Tangent is a Latin word that means 'touching', and that is what this line does to the circle, at point B.

The x coordinate of the point A where the tangent line meets the x axis, is called the secant of θ (we are assuming that the origin is at the center of the unit circle.) Secant is a Latin word meaning 'cutting' which is what this line does to the circle.

They also defined the complement of an angle that is less than a right angle to be the difference between a right angle and it. This got them to define the cosine, cotangent and cosecant as the sine, tangent and secant of the complement of the original angle.

Fortunately for us, all of these six functions are easily related to the sine function, which means that we need only really become familiar with the sine, and we can then figure out what the others are.

FUNCTIONS AND RELATION WITH SINE

Here are the relations between these functions, all of which follow from the definitions from the fact that corresponding angles of similar triangles are equal.

By definition, COS θ= sin(π/2−θ)

From triangle BCD in which the hypotenuse is tanθ and the side not opposite the tanθ is sinθ we get

(cosθ)( tanθ) = sinθ

which means

tanθ = sinθ/​ cosθ =sinθ​/ sin(π/2−θ)

The complementary version of this is:

cotθ= cosθ/sinθ​=sin(π/2−θ)/sinθ

From the triangle 

we similarly get

 (secθ)(sinθ)=tanθ

which means

secθ=1/cosθ​

and the complementary version is

cscθ=1/sinθ​

So all this explains why every ratio of side lengths of a right triangle has a name of its own.

BASIC THEOREMS OF TRIGONOMETRY

 

1.The Pythagorean Theorem:

This famous result states that the square of the hypotenuse of a right triangle is the sum of the squares of its other two sides. Translated to our definitions it says that for any angle, we have

(sinθ) ^2+(cosθ) ^2=1,

which implies that, up to sign we have   

cosθ = sqrt{1-( sinθ)^2}

2. The Law of Sines:

This states that in any triangle ABC the ratio of the sines of its angle at A to its angle at B is the ratio of the lengths of the side opposite A to the side opposite B. If we describe these lengths as l(BC) and l(AC) respectively, we have

sinA/sinB ​= ∣BC∣/∣AC∣​

3. The Law of Cosines:

 This statement gives the length of the side BC of  triangle in terms of the lengths of AB and AC and its angle at A

|BC|^2 = |AB|^2 + |AC|^2 - 2 |AB||AC| cosA

 

CONCLUSION

The more progressive use of trigonometry in a student’s life is to analyze and manipulate equations using trigonometric functions, such as sine, cosine and tangent, and algebra.

Trigonometric functions can be used in obtaining unknown angles and distances from known or measured angles in geometric figures. Trigonometry was developed from a need to compute angles and distances in such fields as astronomy, mapmaking, surveying, and artillery range finding and now it is being used in our day to day life.

 

• FAQs

Question: What are the Basics of Trigonometry?

Answer: Trigonometry basics deal with the measurement of angles and problems related to angles. There are six basic trigonometric ratios: sine, cosine, tangent, cosecant, secant and cotangent. All the important concepts covered under trigonometry are based on these trigonometric ratios or functions.

 

Question: Who Invented Trigonometry?

Answer: Hipparchus(c. 190–120 BCE), also known as the "father of trigonometry", was the first to construct a table of values for a trigonometric function.