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Trigonometry Basics | Trigonometry Formulas | sin θ, cos θ, tan θ, cosec θ, sec θ, cot θ

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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.

height-of-statue-using-trigonometry

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°\angle B = 90 \degree as shown in the figure below

trigonometry-terminology-with-triangle-abc

Proof: A\angle A and C\angle C are acute angle triangles

We know that the sum of all the angles in a triangle is 180°180 \degree. Hence in our example of ABC\triangle ABC we can say that,

A+B+C=180°\angle A + \angle B + \angle C = 180 \degree

In ABC\triangle ABC, B\angle B is 90°90 \degree, so we can solve the above statement as:

A+90°+C=180°A+C=180°90°A+C=90°\begin{aligned} \angle A + 90 \degree + \angle C &= 180 \degree \\ \angle A + \angle C &= 180 \degree - 90 \degree \\ \angle A + \angle C &= 90 \degree \\ \end{aligned}

Since the addition of A\angle A and B\angle B is equal to 90°90 \degree. Both these angles are less than 90°90 \degree. Therefore, A\angle A and C\angle C are acute angles.

What are acute angles?

If the measure of an angle is less than 90°90 \degree, that angle is known as an acute angle.


Let us start with C\angle C, as shown in the figure

opposite-and-adjacent-side-to-angle-C-with-hypotenuse

AB=Opposite side to C andBC=Adjacent side to CAC=Hypotenuse of the right angle triangle\begin{aligned} \overline{AB} &= \text{Opposite side to } \angle C \text{ and} \\ \overline{BC} &= \text{Adjacent side to } \angle C \\ \overline{AC} &= \text{Hypotenuse of the right angle triangle} \end{aligned}

Remember: Hypotenuse is the longest side of a right angle triangle.


Similarly for A\angle A, as shown in the figure

Remember: The terminology for Trigonometry changes with respect to angles.

opposite-and-adjacent-side-to-angle-A-with-hypotenuse

BC=Opposite side to A andAB=Adjacent side to AAC=Hypotenuse of the right angle triangle(hypotenuse will still remain the same)\begin{aligned} \overline{BC} &= \text{Opposite side to } \angle A \text{ and} \\ \overline{AB} &= \text{Adjacent side to } \angle A \\ \overline{AC} &= \text{Hypotenuse of the right angle triangle} \\ & \quad \text{(hypotenuse will still remain the same)} \end{aligned}

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\angle A

Ratios Abbreviations
sine of A\angle A sinA\sin A
co-sine of A\angle A cosA\cos A
tangent of A\angle A tanA\tan A
co-tangent of A\angle A cotA\cot A
secant of A\angle A secA\sec A
co-secant of A\angle A cosecA\cosec A

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\sin A and cosA\cos A based on the trigonometric ratios that we learned above

Remember:

Formulae for sinA\sin A:


sinA=Opposite side of AHypotenuse\sin A = {\text {Opposite side of } \angle A \over Hypotenuse}



Formulae for cosA\cos A:


cosA=Adjacent side of AHypotenuse\cos A = {\text {Adjacent side of } \angle A \over Hypotenuse}

Using these formulas:

sinA=Opposite side of AHypotenuse=BCACcosA=Adjacent side of AHypotenuse=ABAC\begin{aligned} \sin A &= {\text {Opposite side of } \angle A \over Hypotenuse} \\ \\ &= {BC \over AC} \\ \\ \\ \cos A &= {\text {Adjacent side of } \angle A \over Hypotenuse} \\ \\ &= {AB \over AC} \\ \end{aligned}

Remember: sin\sin and cos\cos are fundamental trigonometric ratios and the others are derived ratios.

sin-and-cos-of-angle-A

We have now used all the sides of triangle ABC to derive the values of sinA\sin A and cosA\cos A. In a similar manner, you can now derive the sin\sin and cos\cos for C\angle C.

Deriving the values of other trigonometric ratios from sine and co-sine

Tangent of A\angle A

tanA=sinAcosA=Opposite side of AhypotenuseAdjacent side of Ahypotenuse=Opposite side of AhypotenuseAdjacent side of AhypotenusetanA=Opposite side of AAdjacent side of A=BCAB\begin{aligned} \tan A &= {\sin A \over \cos A} \\ \\ &= {{\text{Opposite side of } \angle A \over hypotenuse} \over {\text{Adjacent side of } \angle A \over hypotenuse}} \\ \\ &= {{\text{Opposite side of } \angle A \over \cancel{hypotenuse}} \over {\text{Adjacent side of } \angle A \over \cancel{hypotenuse}}} \\ \\ \tan A &= {\text{Opposite side of } \angle A \over \text{Adjacent side of } \angle A} \\ \\ &= {BC \over AB} \end{aligned}

Co-Tangent of A\angle A

cotA=cosAsinA=Adjacent side of AhypotenuseOpposite side of AhypotenusecotA=Adjacent side of AOpposite side of A=ABBC\begin{aligned} \cot A &= {\cos A \over \sin A} \\ \\ &= {{\text{Adjacent side of } \angle A \over \cancel{hypotenuse}} \over {\text{Opposite side of } \angle A \over \cancel{hypotenuse}}} \\ \\ \cot A &= {\text{Adjacent side of } \angle A \over \text{Opposite side of } \angle A} \\ \\ &= {AB \over BC} \end{aligned}

Secant of A\angle A

secA=1cosA=1Adjacent side of AHypotenusesecA=HypotenuseAdjacent side of A=ACAB\begin{aligned} \sec A &= {1 \over \cos A} \\ &= {1 \over {\text {Adjacent side of } \angle A \over Hypotenuse}} \\ \\ \sec A &= {Hypotenuse \over \text {Adjacent side of } \angle A} \\ \\ &= {AC \over AB} \end{aligned}

Co-Secant of A\angle A

cosecA=1sinA=1Opposite side of AHypotenusesecA=HypotenuseOpposite side of A=ACBC\begin{aligned} \cosec A &= {1 \over \sin A} \\ &= {1 \over {\text {Opposite side of } \angle A \over Hypotenuse}} \\ \\ \sec A &= {Hypotenuse \over \text {Opposite side of } \angle A} \\ \\ &= {AC \over BC} \end{aligned}

Since you now know how to calculate the trigonometric ratios, here's a tip for remembering these formulae:

  1. We have sin\sin and cos\cos as the fundamental trigonometric ratios

  2. If you know the formulae of sin\sin and cos\cos you can easily derive others

  3. tan is the ratio of sin\sin and cos\cos and cot\cot is the reciprocal of tan\tan

  4. sec\sec is the reciprocal of cos\cos

  5. And cosec\cosec is the reciprocal of sin\sin

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