In this notes we'll cover topics about Trigonometry. We'll study about what is Trigonometry, why one should study Trigonometry and then about the basic formulae of Trigonometric Ratios.
Trigonometry is the study of triangles. More specifically right angled triangles.
Why Trigonometry?
To understand why one should use Trigonometry, it is important to understand where you can use it or apply it. Let us imagine that you are visiting the Statue of Unity (world's tallest Statue, situated in Gujarat-India).
You want to know the height of this statue by standing at a distance from it without actually measuring the height of it. We can visualize a right angled triangle being formed by a ray of sight from your eye to the top of the statue (this forms the hypotenuse), the distance between you and the statue (as one side of the triangle) and the line representing the height of the statue (as the third side of the triangle). This visualization is shown in the image below.
Applying some basic trigonometry formulas it becomes easy to calculate the height of the statue.
What is Trigonometry?
Trigonometry consists of three words - tri, gon and metron. Tri means three, gon means sides and metorn means measure. Hence we can say that Trigonometry is the study of relationships between the sides and angles of a triangle, more specifically right angle triangle.
Trigonometry terminology
Consider a right angle triangle ABC, where ∠B=90° as shown in the figure below
Proof: ∠A and ∠C are acute angle triangles
We know that the sum of all the angles in a triangle is 180°. Hence in our example of △ABC we can say that,
∠A+∠B+∠C=180°
In △ABC, ∠B is 90°, so we can solve the above statement as:
∠A+90°+∠C∠A+∠C∠A+∠C=180°=180°−90°=90°
Since the addition of ∠A and ∠B is equal to 90°. Both these angles are less than 90°. Therefore, ∠A and ∠C are acute angles.
What are acute angles?
If the measure of an angle is less than 90°, that angle is known as an acute angle.
Let us start with ∠C, as shown in the figure
ABBCAC=Opposite side to ∠C and=Adjacent side to ∠C=Hypotenuse of the right angle triangle
Remember: Hypotenuse is the longest side of a right angle triangle.
Similarly for ∠A, as shown in the figure
Remember: The terminology for Trigonometry changes with respect to angles.
BCABAC=Opposite side to ∠A and=Adjacent side to ∠A=Hypotenuse of the right angle triangle(hypotenuse will still remain the same)
Trigonometric Ratios
Now let us learn about the different trigonometric ratios with their abbreviations. For explanation purpose let's consider these ratios with respect to ∠A
Ratios
Abbreviations
sine of ∠A
sinA
co-sine of ∠A
cosA
tangent of ∠A
tanA
co-tangent of ∠A
cotA
secant of ∠A
secA
co-secant of ∠A
cosecA
Note: "co" before the ratios stands for the Complementary of that specific ratio.
Deriving the values of sine and co-sine
Let us now derive the values of sinA and cosA based on the trigonometric ratios that we learned above
Remember:
Formulae for sinA:
sinA=HypotenuseOpposite side of ∠A
Formulae for cosA:
cosA=HypotenuseAdjacent side of ∠A
Using these formulas:
sinAcosA=HypotenuseOpposite side of ∠A=ACBC=HypotenuseAdjacent side of ∠A=ACAB
Remember: sin and cos are fundamental trigonometric ratios and the others are derived ratios.
We have now used all the sides of triangle ABC to derive the values of sinA and cosA. In a similar manner, you can now derive the sin and cos for ∠C.
Deriving the values of other trigonometric ratios from sine and co-sine
Tangent of ∠A
tanAtanA=cosAsinA=hypotenuseAdjacent side of ∠AhypotenuseOpposite side of ∠A=hypotenuseAdjacent side of ∠AhypotenuseOpposite side of ∠A=Adjacent side of ∠AOpposite side of ∠A=ABBC
Co-Tangent of ∠A
cotAcotA=sinAcosA=hypotenuseOpposite side of ∠AhypotenuseAdjacent side of ∠A=Opposite side of ∠AAdjacent side of ∠A=BCAB
### Secant of $\angle A$
secAsecA=cosA1=HypotenuseAdjacent side of ∠A1=Adjacent side of ∠AHypotenuse=ABAC
### Co-Secant of $\angle A$
cosecAsecA=sinA1=HypotenuseOpposite side of ∠A1=Opposite side of ∠AHypotenuse=BCAC
Since you now know how to calculate the trigonometric ratios, here's a tip for remembering these formulae:
We have sin and cos as the fundamental trigonometric ratios
If you know the formulae of sin and cos you can easily derive others
tan is the ratio of sin and cos and cot is the reciprocal of tan
sec is the reciprocal of cos
And cosec is the reciprocal of sin
Suggested Notes:
tan (A + B) = √3 and tan (A -B) = 1/√3 | Trigonometry Numerical
15 cot A = 8 | Find the value of sin A and sec A | Trigonometry Numerical
PQ = 12cm and PR = 13cm | Find tan P - cot R | Trigonometry Numerical
cot θ = 7/8 | Find all other trigonometric ratios | Trigonometry Numerical
Suggested Notes:
tan (A + B) = √3 and tan (A -B) = 1/√3 | Trigonometry Numerical
15 cot A = 8 | Find the value of sin A and sec A | Trigonometry Numerical
PQ = 12cm and PR = 13cm | Find tan P - cot R | Trigonometry Numerical
cot θ = 7/8 | Find all other trigonometric ratios | Trigonometry Numerical
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