Trigonometry
Trigonometry is a branch of mathematics that studies the relationship between side lengths and the angle of triangles. The word trigonometry has Greek origin from Greek trigonal which means “triangle” and metron which means “measure”.
Trigonometry as a discipline has found its applications in astronomy, navigation, surveying, periodic functions, optics, acoustics, electronics, biology and many more. So, let’s learn about trigonometric table of all angles.
Right-angled triangle
Before we dig down and try to understand trigonometric functions; let’s first understand the right-angled triangle; its sides and angles and how we can utilize them in trigonometry.
click here – Cross browser testing checklist before going live
As the name signifies, a right-angled triangle is a triangle in which one of the angles is a right angle. The above-mentioned triangle is right-angled at “C”.
Since in any given triangle all interior angles add up to 180°, hence in a right-angled triangle; the other two angles in a right triangle add up to 90°. These two angles which add up to 90° are known as complementary angles.
Now, the side opposite to the right angle which is, in this case, is “c” is known as hypotenuse which is also the longest side in a right-angled triangle. The sides which are adjacent to the right angle are called legs and are denoted by “a” and “b” in the above figure.
Trigonometry ratio
A trigonometric ratio, as the name suggests, is the ratio between edges of a right-angled triangle and this ratio relates an acute angle of a right-angled triangle to the ratio of two sides. In the above figure is the right angle at “C”; side “c” represent hypotenuse; side “b” represent adjacent (base) side and side “a” represent perpendicular side (opposite).
Now, there are six major trigonometric functions that are widely used in mathematics as mentioned below:
- Sine
- Cosine
- Tangent
- Cosecant (Reciprocal of sine function)
- Secant (Reciprocal of cosine function)
- Cotangent (Reciprocal of tangent function)
These functions have a well-defined ratio which is shown as below:
| Functions | Ratio |
| Sine | The ratio of the opposite side (perpendicular) to that of the hypotenuse |
| Cosine | The ratio of the adjacent side (base) to that of the hypotenuse |
| Tangent | The ratio of the opposite side (perpendicular) to that of the adjacent (base) |
| Cosecant | The inverse of sine function i.e ratio of hypotenuse side to that of the opposite (perpendicular) |
| Secant | The inverse of cosine function i.e ratio of hypotenuse side to that of adjacent |
| Cotangent | The inverse of tangent function i.e ratio of adjacent side (base) to that of the opposite (perpendicular) |
From the above information, it is now easy to determine the trigonometric function for an acute angle “A” and is as mentioned below:
| Functions | Short form | Ratio |
| Sine | sin A | a/c |
| Cosine | cos A | b/c |
| Tangent | tan A | a/b |
| Cosecant | cosec A | c/a |
| Secant | sec A | c/b |
| Cotangent | cot A | b/a |
Moreover, we read at the beginning of the topic that in a right-angled triangle; the sum of the remaining two acute angles add up to 90° which means both these acute angles are a complement to each other.
This information can be further utilised to derive the relationship between trigonometric functions as shown below:
| Functions | Short form | Relationship using degrees |
| Sine | sin A | Sin α = Cos (90°- α) |
| Cosine | cos A | Cos α = Sin (90°- α) |
| Tangent | tan A | Tan α = Cot (90°- α) = Sin α/ Cos α |
| Cosecant | cosec A | Cosec α = Sec (90°- α) |
| Secant | sec A | Sec α = Cosec (90°- α) |
| Cotangent | cot A | Cot α = Tan (90°- α) = Cos α/ Sin α |
| Let α represent an acute angle. | ||
*From the above table we can conclude that if ∠A and ∠B are acute angles of a right triangle; then Sin A= Cos B which means sine of any acute angle is equal to the cosine of its complement and cosine of any acute angle is equal to the sine of its complement.
click here – GC Full Form & Meaning
Examples
Example 1: If Cos α is ¾ then what is the value of Sec α?
Solution: As we know; Sec α = 1/Cos α.
Hence Sec α = 1/ (3/4)
Sec α = 4/3
Example 2: If Sin α = ½ and Cos α= ¾; then what is the value of Tan α?
Solution: As we know; Tan α = Sin α/Cos α
Hence Tan α = (1/2)/(3/4)
Tan α = 2/3
Trigonometric Table Of All Angles
Also, a standard trigonometric table of all angles is available which can be utilized for calculations and is as shown below:
| Angles (In degrees) | 0° | 30° | 45° | 60° | 90° | 180° | 270° | 360° |
| Sin | 0 | ½ | 1/√2 | √3/2 | 1 | 0 | -1 | 0 |
| Cos | 1 | √3/2 | 1/√2 | ½ | 0 | -1 | 0 | 1 |
| Tan | 0 | 1/√3 | 1 | √3 | Not defined | 0 | Not defined | 1 |
| Cot | Not defined | √3 | 1 | 1/√3 | 0 | Not defined | 0 | Not defined |
| Cosec | Not defined | 2 | √2 | 2/√3 | 1 | Not defined | -1 | Not defined |
| Sec | 1 | 2/√3 | √2 | 2 | Not defined | -1 | Not defined | -1 |
Examples.
Example 1:
The value of Sin 30° is ½ whereas the value of Cot 270° is 0.
Example 2:
If Cos α =1/2, then what is the value of α?
Solution: From the above table, we can see that Cos 60° = ½, hence the value of α is 60°.
Example 3:
If sin (2x+10) ° = Cos (x+20) °, then find the value of x?
Solution: We know that in a right-angled triangle; acute angles are complementary.
Therefore; 2x+10+x=20=90
3x+30=90
3x=60
x=20
Conclusion
Trigonometry is a mathematical concept that you can see in use in everyday life. It is used to calculate the height of buildings, in architecture and in optical science. Knowing your trigonometry basics like ratio values and formulas makes the work extremely easy for professionals.