ANGLE: Angle is defined as the measure of the rotation of a ray.

RADIAN: Radian is defined as the angle subtended by the arc of a circle at its centre.

At the centre, a radian = 180 degrees

Trigonometric Functions

We mainly find six types of trigonometric functions that can have input as an angle value of a right triangle. We can consider it as the domain of this function, and output from these is a numeric value that can be considered the range of these functions.

Any trigonometric function like f(x) = sinθ has a domain that can be given by tiers or radians, and its value can vary within a range for sinθ; it is a number ranging [-1, 1]. We may find the usage of Trigonometric features in calculus, geometry, and algebra.

Generally, These functions are trigonometric ratios. Named as sine, cosine, secant, cosecant, cotangent, and tangent. The trigonometric functions can be explained by the ratio of sides of a right-angled triangle. 

A right-angled triangle has sides named as base, perpendicular, and hypotenuse. Using these parameters, we can calculate and approximate the values of sine, cosine, tangent, secant, cosecant, and cotangent trigonometric functions.

Trigonometric Functions Formulas:

Sinθ= Perpendicular/Hypotenuse

Cosθ
= Base/Hypotenuse

Tanθ
= Perpendicular/Base

Secθ
= Hypotenuse/Base

Cosecθ
= Hypotenuse/Perpendicular

Cotθ
=Base/ Perpendicular 

Trignometric Function Values

Trigonometric Functions in Four Quadrants

θ is an acute angle (θ < 90°) that is measured with reference to the x-axis in the anticlockwise direction. Furthermore, those trigonometric functions are positive or negative (+ or -) within the different quadrants. The trigonometric functions of Sinθ, Cosecθ are advantageous in quadrants I and II and are terrible in quadrants III and IV. 

The trigonometric functions have values of θ (90° – θ) inside the first quadrant. The cofunction identities provide the interrelationship among the one-of-a-kind complementary trigonometric features for the perspective (90° – θ).

sin(90°−θ) = cos θ

cos(90°−θ) = sin θ

tan(90°−θ) = cot θ

cot(90°−θ) = tan θ

sec(90°−θ) = cosec θ

cosec(90°−θ) = sec θ

The trigonometric functions will interchange with its complement when the angle is represented in the form of 90nθ, where n is an odd integer. In contrast, the functions will remain the same when the angle is written as 180nθ, where n is an integer.

The complements are as follows:

Sinθ⇔Cosθ, Tanθ⇔Cotθ, Secθ⇔Cosecθ. 

Domain and Range of Trigonometric Functions

Representation of the area of any trigonometric can be done using the value of θ, where the calculated area can be considered as the variety of the trigonometric function. The domain values of these trigonometric functions can be represented using the tiers or radians. Talking about the range of these functions, this can be represented by quantitative values like price. Generally, the domain of the trigonometric function gets converted into an actual range value. The trigonometric functions also have periodic capabilities. The below table represents the domain and range of these trigonometric functions.

Trignometric Function Graphs

Cosec x

Tan x

Cos x

Sin x

Cot x

Sec x

Trignometric Function Identities

Reciprocal Identities

For simplification, these identities are used.

• cosec θ = 1/sin θ

• sec θ = 1/cos θ

• cot θ = 1/tan θ

• sin θ = 1/cosec θ

• cos θ = 1/sec θ

• tan θ = 1/cot θ

Pythagorean Identities

• Sin2θ + Cos2θ = 1

• 1 + Tan2θ = Sec2θ

• 1 + Cot2θ = Cosec2θ

• sin(x+y) = sin(x)cos(y) + cos(x)sin(y)

• cos(x+y) = cos(x)cos(y) – sin(x)sin(y)

• tan(x+y) = (tan x + tan y)/ (1−tan x tan y)

• sin(x–y) = sin(x)cos(y) – cos(x)sin(y)

• cos(x–y) = cos(x)cos(y) + sin(x)sin(y)

• tan(x−y) = (tan x–tan y)/ (1+tan x tan y)

Half-angle identities

• sin A/2 = ±√[(1 – cos A) / 2]

• cos A/2 = ±√[(1 + cos A) / 2]

• tan A/2 = ±√[(1 – cos A) / (1 + cos A)] (or) sin A / (1 + cos A) (or) (1 – cos A) / sin A

Double Angle

• sin(2x) = 2sin(x) cos(x) = [2tan x/(1+tan2 x)]

• cos(2x) = cos2(x)–sin2(x) = [(1-tan2 x)/(1+tan2 x)]

• cos(2x) = 2cos2(x)−1 = 1–2sin2(x)

• tan(2x) = [2tan(x)]/ [1−tan2(x)]

• cot(2x) = [cot2(x) – 1]/[2cot(x)]

• sec (2x) = sec2 x/(2-sec2 x)

• cosec (2x) = (sec x. cosec x)/2

Triple Angle

• Sin 3x = 3 sin x – 4sin3x

• Cos 3x = 4cos3x – 3cos x

• Tan 3x = [3tanx-tan3x]/[1-3tan2x]

Product Identities

• 2sinx⋅cosy=sin(x+y)+sin(x−y)

• 2 cosx⋅cosy=cos(x+y)+cos(x−y)

• 2 sinx⋅sinx=cos(x−y)−cos(x+y)

Sum of Identities

• sinx+siny=2sin((x+y)/2) . cos((x−y)/2)

• sinx−siny=2 cos((x+y)/2) . sin((x−y)/2)

• cosx+cosy=2cos((x+y)/2) . cos((x−y)/2)

• cosx−cosy=−2sin((x+y)/2 . sin((x−y)/2)

Some Negative Angles

• sin(-θ) = – sinθ

• cos (-θ) = cosθ

• tan (-θ) = – tanθ

• cot (-θ) = – cotθ

• sec (-θ) = secθ

• cosec (-θ) = – cosecθ

Inverse Trignometric Identities

Trigonometric functions are invertible functions. So, Sin θ = x can be written as sin-1x = θ. In the inverse functions, the domain and range will interchange with the original function. Some of the properties of inverse trigonometric functions is written as:

• Sin-1(-x) = -Sin-1x

• Tan-1(-x) = -Tan-1x

• Cosec-1(-x) = -Cosec-1x

• Cos-1(-x) = π – Cos-1x

• Sec-1(-x) = π – Sec-1x

• Cot-1(-x) = π – Cot-1x

Trigonometric Equations

An equation that entails trigonometric capabilities of unknown angles is called a trigonometric equation.

Solution of a Trigonometric Equation

An answer to a trigonometric equation is the value of the unknown angle that satisfies the equation.

A trigonometric equation may additionally have a countless range of answers.

Principal Solution

The solutions of a trigonometric equation for which 0 ≤ x ≤ 2π are called principal solutions.

A solution of a trigonometric equation involving ‘n’, which gives all solutions of a trigonometric equation, is called the general solution.

Basic Rules of Triangles

In a triangle ABC, let the angles be A, B, and C, and the opposite sides are a, b, and c, respectively.

Sine rule

sin Aa=sin Bb=sin Cc

Cosine rule

a2=b2+c2-2bc cos A

b2=c2+a2-2ac cos B

c2=a2+b2-2ab cos C

Integration of Trigonometric function

The integration of trigonometric functions represents the area under the curve with respect to the x-axis if we are integrating it with respect to x.

Some of the integration formulas are given below:

∫ cosx dx = sinx + C

∫ sinx dx = -cosx + C

∫ sec2x dx = tanx + C

∫ cosec2x dx = -cotx + C

∫ secx.tanx dx = secx + C

∫ cosecx.cotx dx = -cosecx + C

∫ tanx dx = log|secx| + C

∫ cotx.dx = log|sinx| + C

∫ secx dx = log|secx + tanx| + C

∫ cosecx.dx = log|cosecx – cotx| + C

Solved Examples on Trigonometric Function

Example: Estimate the value of sin 75°.

Solution :

Here we can see that the value of angle is 75°

In the above formulas we can find that 

sin(A+B) = sinA.cosB + cosAsinB

And we can write 75° as (30° + 45°)

So 

Sin75° = sin(30° + 75°)

Now we can think of sin(30° + 45°) as sin(A+B)

So 

Sin75° = sin(30° + 75°) = sin 30°. Cos45° + cos30°.sin 45°

Using the above table, we can find the values in numerical form.

sin 75° = (½)(1/√2) + (√3/2)(1/√2)

 = (√3 + 1)/2√2

Answer: The value of sin 75° is (√3 + 1)/2√2

Conclusion

Trigonometry might not have many regular applications, but it does assist in working with triangles. It is a beneficial complement to geometry and real measurements and, as such, nicely worth growing information of the fundamentals, even in case you by no means want to progress similarly.