Trigonometric Identities
Trigonometric identities are math’s equations that are true for all values of the variables in the equation. They use trigonometric functions. We can use these identities to simplify trigonometric expressions, solve trigonometric equations, and prove other mathematical theorems.
All Trigonometric Identities
All Trigonometric Identities are given in the below figure:
Derivation of Pythagorean Identities
For any real number θ, we shall derive the following three fundamental identities;
- sin²θ + cos²θ = 1
- tan2θ + 1 = sec2θ
- cot2θ + 1 = cosec2θ
Derivation of sin²θ + cos²θ = 1
Consider an angle <XOP = θ in the standard position. Take a point P on the terminal line of the angle θ. Draw PQ perpendicular to P on OX.
From the above figure, OPQ is a right-angled triangle.
By Pythagoras theorem
(OP)² = (OQ)² + (PQ)²
OR
z² = x² + y²
Dividing both sides by z²
then;
z²/ z² = x² / z² + y²/z²
1= (x/z)2 + (y/z)2
1= cos²θ + sin²θ
OR
sin²θ + cos²θ = 1
Derivation of tan2θ + 1 = sec2θ
Consider an angle <XOP = θ in the standard position. Take a point P on the terminal line of the angle θ. Draw PQ perpendicular to P on OX.
From the above figure, OPQ is a right-angled triangle.
By Pythagoras theorem
(OP)² = (OQ)² + (PQ)²
OR
z² = x² + y²
Dividing both sides by x²
then;
z²/ x² = x² /x² + y²/x²
(z/x)2 = 1+ (y/x)2
sec2θ = 1 + tan2θ
OR
tan2θ + 1 = sec2θ
Derivation of cot2θ + 1 = cosec2θ
Consider an angle <XOP = θ in the standard position. Take a point P on the terminal line of the angle θ. Draw PQ perpendicular to P on OX.
From the above figure, OPQ is a right-angled triangle.
By Pythagoras theorem
(OP)² = (OQ)² + (PQ)²
OR
z² = x² + y²
Dividing both sides by y²
then;
z²/ y² = x² /y² + y²/y²
(z/y)2 = (x/y)2 + 1
cosec2θ = cot2θ + 1
OR
cot2θ + 1 = cosec2θ
Examples with Solutions
Example 1
Prove that (Sin x/Cosec x) + (Cos x/sec x) = 1
SOLUTION
L.H.S = (Sin x/Cosec x) + (Cos x/sec x)
= Sin x (1/Cosec x) + Cos x (1/Sec x)
Where; 1/Cosec x = Sin x and 1/Sec x = Cos x
So,
= Sin x . Sin x + Cos x . Cos x
= Sin2 x . Sin2 x + Cos2 x . Cos2 x
= 1
= R.H.S
Example 2
Prove that (Sec x – Cos x)/(1+ Cos x) = Sec x-1
L.H.S = (Sec x – Cos x)/(1+ Cos x)
= (1/ Cos x) – Cos x / 1+ Cos x
= (1 – Cos2 x/ Cos x) / (1 + Cos x) = (1 – Cos2 x) / Cos x (1+ Cos x)
= (1- Cos x) (1+ Cos x) / Cos x (1+ Cos x)
= (1- Cos x) / (Cos x) = (1/Cos x) – (Cos x / Cos x)
= Sec x-1
=R.H.S
Measurement of Angles in Trigonometry
The amount of rotations determines the size of any angle. In trigonometry, two systems of measuring angles are used in trigonometry.
- Sexagesimal or English system (Degree)
- Circular measure system (Radian)
Sexagesimal or English System (Degree) in Trigonometry
The sexagesimal system is older and is more commonly used. The name derives from the Latin for “sixty.” The fundamental unit of angle measure in the sexagesimal system is the degree of arc. By definition, when a circle is divided into 360 equal parts, then
- One degree = 1/360th part of a circle.
- Therefore, one full circle = 360 degrees.
- The symbol of degrees is denoted by ()º.
- Thus an angle of 20 degrees may be written as 20°.
- Since there are four right angles in a complete circle.
- One right angle = 1/4 circle = 1/4(360°) = 90°
The degree is further subdivided in two ways, depending on whether we work in the standard or decimal sexagesimal systems. In the standard sexagesimal system, the degree is subdivided into 60 equal parts, called minutes, denoted by the symbol (‘), and the minute is further subdivided into 60 similar pieces, called second and indicated by the symbol (“).
Therefore
- 1 minute = 60 seconds
- 1 degree = 60 minutes = 3600 seconds
- 1 circle = 360 degrees = 21600 minutes = 12.96,000 sec.
In the decimal sexagesimal system, angles smaller than 1° are expressed as decimal fractions of a degree. Thus one-tenth (1/10) of a .degree is expressed as 0.1 in the decimal sexagesimal system and as 6′ in the common sexagesimal system; one-hundredth (1/100) of a degree is 0.01° in the decimal system, and 36″ in the common system; and 47(1/9) degrees comes out (47.111 …)° in the decimal system and 47°6‘40″ in the common system.
Circular measure system (Radian) in Trigonometry
This system is comparatively recent. The unit used in this system is called a Radian.
The Radian is defined as:
The angle subtended at the center of a circle by an arc equal in length to the circle’s radius.
Radian
As shown in the figure;
Arc AB is equal in length to the radius OB of the circle. The subtended, <AOB is then one Radian.
i.e. m <AOB = 1 radian.
Relation between Degree and Radian Measure
Consider a circle of radius r. then the circumference of the circle is 2πr.
By definition of radian:
An arc of length ‘r’ subtends an angle = 1 radian
Hence. An area of length 2πr subtends an angle = 2π radian
Also an arc of length 2πr subtends an angle = 360°
Then;
2π radians = 360°
OR
π radians = 180º
1 radian = 180º/ π
1 radians = 180º/ 3.1416
1 radians = 57.3°
Therefore to convert radians into a degree.
we multiply the number of radians by 180º/ π or 57.3°
Now Again
360° = 2π radians
1° = 2πr/ 360° radians
1° = π/ 180°
1° = 3.146/180°
1° = 0.001745 radians
Therefore, to convert degrees into radians. We multiply the number of degrees by π/180 or 0.0175.
Note: One complete revolution = 360° = 27 radii.
Relation b/w Length of Circular Arc & the Radian Measure of its Central Angles
Let “l” be the length of a circular area, AB of a circle of radius r, and θ be its central angle measure in radians. Then the ratio of l to the circumference 2πr of the circle is the same as the ratio of θ to 2π.
Therefore
l : 2πr = θ : 2π
l/2πr = θ / 2π
l / r = θ
l = θ r
Where θ is in Radian.
Note:
If the angle is given in degree measure, we must convert it into a Radian measure before applying the formula.
FAQ’s
Why do students find trigonometry difficult?
The relationships between the sides and angles of triangles are the subject of the mathematical discipline of trigonometry. Trigonometry is difficult for many students for several reasons, including:
New concepts
Abstract nature
Complex calculations
Lack of practice
Lack of foundational
What is the difference between Degree and Radian?
In trigonometry, degree, and radians are units used to measure angles.
Degree: A degree is a unit of angle measurement representing 1/360th of a full rotation. It is denoted by the symbol °. A complete rotation of 360 degrees takes us back to the starting point.
Radian: A radian is a unit of angle measurement defined as the angle subtended by an arc of length equal to the circle’s radius. One radian is equivalent to 180/π degrees or approximately 57.3 degrees. Radians are denoted by the symbol “rad.”
Is trigonometry used in physics?
Yes, trigonometry is commonly used in physics. Trigonometry is the branch of mathematics that studies the relationships between triangle sides and angles. Triangles frequently appear in physics when looking at the motion and forces of objects.
Trigonometry determines the angle at which an object is launched, the height it reaches, the distance it travels, and the time it takes to get to a certain point when studying projectile motion, which is the motion of an object through the air.
What is the formula for cos θ?
Yes, trigonometry is commonly used in physics. Trigonometry is the branch of mathematics that studies the relationships between triangle sides and angles. Triangles frequently appear in physics when looking at the motion and forces of objects.
In trigonometry, this formula determines the co-sine of an angle in a right triangle. The adjacent side is adjacent to the angle, and the hypotenuse is the longest side of the triangle opposite the right angle.
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