Clear Understanding on Sin, Cos and Tan (Trigonometric Functions)

 

Clear Understanding on Sin, Cos and Tan (Trigonometric Functions)

Trigonometry (from Ancient Greek τρίγωνον (trígōnon) 'triangle', and μέτρον (métron) 'measure') is a branch of mathematics concerned with relationships between angles and side lengths of triangles. 

All trigonometric formulas are divided into two major systems:

• Trigonometric Functions
• Trigonometric Identities

In particular, the trigonometric functions  relate the angles of a right triangle with ratios of its side lengths. Also, called as trigonometric rations.

Trigonometric functions are also known as Circular or cyclic Functions.

Trig functions can be simply defined as the functions of an angle of a triangle. It means that the relationship between the angles and sides of a triangle are given by these trig functions.

There are basically 6 ratios used for finding the elements in Trigonometry, those are sine(sin), cosine(cos),  tangent(tan), cotangent(cot), secant(sec) and cosecant(cosec).

Sin, cos and tan are the primary trigonometry functions whereas the other three functions cot, sec and cosec can be obtained from the primary functions. 

Trigonometric Identities are formulas that involve Trigonometric functions. Trigonometric identities can be, 

1. Reciprocal Identities, like cosec θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
2. Periodicity Identities (in Radians), 
3. Cofunction Identities (in Degrees)
4. Sum & Difference Identities
5. Double Angle Identities
6. Triple Angle Identities
7. Half Angle Identities
8. Product identities 
9. Sum to Product Identities
10. Inverse Trigonometry Formulas

The trigonometric identities hold true only for the right-angle triangle. All the fundamental trigonometric identities are derived from the six trigonometric ratios. Using these trigonometric identities or formulas, complex trigonometric questions can be solved quickly.

Trigonometric functions are widely applied in calculus, geometry, algebra and related domains. In trigonometry, the angle measured is represented in radians (refer other blog, why redians is preferred). Trig function in excel takes the angle in radians for which you want the respective sine, cos or tan value. 

If angel is provided as degree, you can use RADIANS function in Excel to convert it to radians (1 radian = 57.2958 degree).

If angle in radians to be converted to degrees, you can use DEGREES function in Excel ( 1 degree = 0.01745 radian).

By using a right-angled triangle as a reference, the trigonometric functions and identities are derived.

Before going into the study of the trigonometric functions we will learn about the 3 sides of a right-angled triangle.

The three sides of a right-angled triangle are as follows,

• Base The side on which the angle θ lies is known as the base. (used as A in below formula, A for Adjacent)
• Perpendicular It is the side opposite to the angle θ  in consideration (used as O in below formula, O for opposite)
• Hypotenuse It is the longest side in a right-angled triangle and opposite to the 90° angle (used as H in below formula).

following are formulae for primary trigonometric functions.

The three trigonometric ratios can be used to determine a missing side or a missing angle.

In Summary,

For a given angle, each ratio stays the same no matter how big or small the triangle is.
Refer link below related Vignettes, in heading Angles from 0 to 360 degree.

Sine, Cosine, Tangent (mathsisfun.com) 

Let’s place the right angle triangle, in unit circle, on the standard cartesian axes for an angles associated with respective quadrant (and it is unit radius circle).

In all of the above plots, Base or adjacent side is X axis. Hypotenuse is radius and constant for all points on the circle.

Equation of a Unit Circle

The general equation of a circle is (x – a)2 + (y – b)2 = r2, which describes a circle having the center (a, b) and the radius r.

This equation can be simplified to represent the equation of a unit circle. 

A unit circle is made with its center at the point(0, 0), which is the origin of the coordinate axes and a radius of 1 unit. 

Hence the equation of the unit circle is (x – 0)2 + (y – 0)2 = 12. 

A simplified version of this equation is shown below.

Equation of a Unit Circle: x2 + y2 = 1

All the points in the circle and in the quadrants, according to the above equation, are satisfied.

Finding Trigonometric Functions Using a Unit Circle

The trigonometric functions sine, cosine, and tangent can be calculated using a unit circle.  Refer fig below,

Consider a right triangle placed within a unit circle in the cartesian coordinate plane.

The radius vector forms an angle θ with the positive x-axis and the coordinates of the endpoint of the radius vector is (x, y).

Right triangles have a base and an altitude, which are represented by the values of x and y respectively.

We now have a right angle triangle that has sides 1, x, and y. 

 We can find the values of the trigonometric ratio by applying this in trigonometry:

• cosθ = Base/Hypotenuse = x/1
• sinθ = Altitude/Hypoteuse = y/1

We now have 

cosθ = x,

sinθ = y, 

Thus, coordinates of radius vector which form angle  θ can be give by as shown in picture.

for other trig functions,

tanθ = sinθ/cosθ =  y/x

secθ = 1/cosθ =1/x

cscθ = 1/sinθ  = 1/y

cot(θ) = cosθ/sinθ = x/y

Lets calculate those for angle of 55 degree.

Cast rule in a Unit Circle

Other perspective, visualization.

Below plots of Trig functions are for unit radius circle.

You should be able to interpret the unit circle to determine the value of $�$.

Refer link below related Vignettes, in heading Angles from 0 to 360 degree.

Sine, Cosine, Tangent (mathsisfun.com) 

The trig functions that possess a domain input value as an angle of a right triangle, and a numeric answer as the range is the basic trigonometric functions definition.

let us see the domain and range of each function, which is to be graphed in XY plane.

Here is the graph for all the functions based on their respective domain and range.

The trig functions are the periodic functions. The smallest periodic cycle is 2π but for tangent and the cotangent it is π.  

Explanation of Trigonometric Functions in Four Quadrants

Other questions related to Unit Circle:

Must a right angled triangle with its points on the circumference of a circle, have a hypotenuse that is the diameter of the circle?  (Circle with inscribed triangle)

Yes, it must have the hypotenuse as the diameter of the circle. Refer below plot, angle ∠ABC is a right angle.

Triangle with inscribed circle

What is the difference between the sides of a right triangle if the point at which the inscribed circle touches the hypotenuse and the hypotenuse divides it into two segments of length one 5 cm and the other 12 cm?

Vignette to visualize and understand all trig ratio in relation to point on circle.

 Helps to understand, the circumference of the circle is the hypotenuse of the triangle. 

Trig unit circle review (article) | Khan Academy

Reference

1. https://byjus.com/maths/trigonometry-formulas/
2. https://www.geeksforgeeks.org/what-are-the-six-trigonometry-functions/
3. https://teachablemath.com/sine-cosine-and-tangent-in-the-four-quadrants/
4. https://thirdspacelearning.com/gcse-maths/geometry-and-measure/sin-cos-tan/
5. https://www.mathsisfun.com/sine-cosine-tangent.html
6. https://www.nagwa.com/en/explainers/869142503292/
7. https://www.quora.com/Must-a-right-angled-triangle-with-its-points-on-the-circumference-of-a-circle-have-a-hypotenuse-that-is-the-diameter-of-the-circle-Could-the-hypotenuse-be-a-chord-randomly-placed-within-the-circle
8. https://www.quora.com/What-is-the-difference-between-the-sides-of-a-right-triangle-if-the-point-at-which-the-inscribed-circle-touches-the-hypotenuse-and-the-hypotenuse-divides-it-into-two-segments-of-length-one-5-cm-and-the-other-12-cm
9. Trig unit circle review (article) | Khan Academy
10. Unit Circle: Chart, Equations, Trigonometric Functions, and Radians - Education Spike
11. https://www.nagwa.com/en/explainers/869142503292/