Key concepts of fundamental trigonometric functions, trigonometric identities, proving other trigonometric identities, simplifying trigonometric expressions, inverse trigonometric expressions, as well as the domain and range of the inverse trigonometric functions.

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Trigonometry:

In mathematics, trigonometry is a concept created by the Greek mathematician Hipparchus. It is considered as one of the branches of the subject that deals with the relationship among the sides, lengths, heights and angles of a triangle. This is notable for finding the unknown values of the sides, lengths, heights, and angles of a right-angled triangle through the application of trigonometric identities, formulas, and functions. While, the angles of the right triangle can be measured through degrees or radians where the commonly used degrees for solving and calculations of the right triangle are 0°, 30°, 45°, 60° and 90°.

Real-life Application of Trigonometry

Trigonometry is used in measuring the heights of mountains, towers, or buildings:

Heights of mountains tall towers can easily be found using trigonometry, if you want to find the height of a tower measuring the horizontal distance from the base of the tower and find the angle of elevation to the top of the tower using a sextant, then you can easily find the height of the tower, mountain or any other thing.

Sub-Topics List:

Trigonometric Functions

Trigonometric Identities

Pythagorean Identities

Sum and Difference identities

Sine Laws

Cosine Laws

Euler’s Formula

Inverse Trigonometric Expressions

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Sub-Topics

Trigonometric Functions

Trigonometric functions, “trigonometric ratios”, are the functions used in the calculations of unknown specific distances and angles of a triangle, especially a right triangle. There are six (6) commonly used functions of an angle given their names and abbreviations, respectively; sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

Functions	Abbreviation	Relationship to sides of a right triangle
Sine Function	sin	Opposite side/ Hypotenuse
Tangent Function	tan	Opposite side / Adjacent side
Cosine Function	cos	Adjacent side / Hypotenuse
Cosecant Function	cosec	Hypotenuse / Opposite side
Secant Function	sec	Hypotenuse / Adjacent side
Cotangent Function	cot	Adjacent side / Opposite side

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Trigonometric Identities

Trigonometric identities, “trigonometric formulas” are the equations used for right–angled triangles. Here are the several trigonometric identities you can apply on identifying unknown parts of the triangle:

sin²θ + cos²θ = 1

tan²θ + 1 = sec²θ

cot²θ + 1 = cosec²θ

Proving the Trigonometric Identities

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Pythagorean Identities

sin²θ + cos²θ = 1
tan2θ + 1 = sec2θ
cot2θ + 1 = cosec2θ
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
cot 2θ = (cot²θ – 1) / 2 cot θ

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Sum and Difference identities

For angles a and b, we have the following relationships:

sin(a + b) = sin(a)cos(a) + cos(a)sin(b)

cos(a + b) = cos(a)cos(a) – sin(a)sin(b)

tan(a + b) = tan (a) + tan (b)/1-tan(a) tan (b)

sin(a – b) = sin(a)cos(b) – cos(a)sin(b)

cos(a –b) = cos(a)cos(b) + sin(a)sin(b)

tan(a-b) = tan (a) + tan (b)/1+ tan(a) tan (b)

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Sine Laws

If A, B and C are angles and a, b and c are the sides of a triangle, then,

a/sinA = b/sinB = c/sinC

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Cosine Laws

c2 = a2 + b2 – 2ab cos C

a2 = b2 + c2 – 2bc cos A

b2 = a2 + c2 – 2ac cos B

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Euler’s Formula

Euler’s formula is eix = cos x + i sin x where x is the angle and i is the imaginary number.

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Inverse Trigonometric Expressions

Inverse trigonometric expressions are expressions that are useful in identifying the value of the angle of a right triangle given only its two sides, whereas it uses inverse trigonometric functions of sine, cosine, ang tangent. These inverse trigonometric functions “undoes” the “does” of the existing trigonometric functions.

The relations of the given inverse trigonometric for sine, cosine, tangent, cosecant, secant, and cotangent are, respectively: arcsine, arccosine, arctangent, arccosecant, arcsecant, and arccotangent. Another way to write x = sin(y) is y = arcsin(x).

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