[Maths Class Notes] on Sec 30 Pdf for Exam

Trigonometric Ratios-

What are Trigonometric Ratios?

Trigonometric ratios in trigonometry are derived from the three sides of a right-angled triangle basically the hypotenuse, the base (adjacent), and the perpendicular which is also known as the opposite.
According to the trigonometric ratio in maths, there are three basic or primary trigonometric ratios also known as trigonometric identities.
To be more specific, they are used in right-angled triangles, the triangles where the value of one angle is equal to 90 degrees.

There are six trigonometric ratios where sine, cosine, and tangent are known as the primary trigonometric functions.

The Primary Trigonometric Functions are Defined as Follows:

NAME	ABBREVIATION	RELATIONSHIP
Sine	Sin	[Sin{text{ }}left( theta right) = {text{ }}frac{{Opposite}}{{Hypotenuse}}]
Cosine	Cos	[Cos{text{ }}left( theta right) = {text{ }}frac{{Adjacent}}{{Hypotenuse}}]
Tangent	Tan	[Tan{text{ }}left( theta right) = {text{ }}frac{{Opposite}}{{Adjacent}}]

The Six Trigonometric Ratios are Defined as Follows:

1. Sine of an angle θ,

[Sin{text{ }}left( theta right) = {text{ }}frac{{Opposite}}{{Hypotenuse}}]

2. Cosine of an angle θ,

[Cos{text{ }}left( theta right) = {text{ }}frac{{Adjacent}}{{Hypotenuse}}]

3. Tangent of an angle θ,

[Tan{text{ }}left( theta right) = {text{ }}frac{{Opposite}}{{Adjacent}}]

4. Cotangent of an angle θ,

[Cot{text{ }}theta {text{ }} = {text{ }}frac{{Adjacent}}{{Opposite}}]

5. Secant of an angle θ,

[Sec{text{ }}theta {text{ }} = {text{ }}frac{{Hypotenuse}}{{Adjacent}}]

6. Cosecant of an angle θ,

[Cosec{text{ }}theta {text{ }} = {text{ }}frac{{Hypotenuse}}{{Opposite}}]

Reciprocal Relations –

[sin{text{ }}left( x right)] can be written as [frac{1}{{Cosecleft( x right)}}]	[cosec{text{ }}left( x right)] can be written as [frac{1}{{Sinleft( x right)}}]
[cos left( x right)]can be written as [frac{1}{{secleft( x right)}}]	[secleft( x right)] can be written as [frac{1}{{Cosleft( x right)}}]
[tanleft( x right)] can be written as [frac{1}{{cotleft( x right)}}]	[cot{text{ }}left( x right)] can be written as [frac{1}{{tanleft( x right)}}]

Before Moving Ahead a Little Information About what a Right-Angled Triangle is?

To understand what a right angle triangle is, let us consider a right-angle triangle named ABC, with its three sides namely the opposite, adjacent, and the hypotenuse. In a right-angled triangle, we generally refer to the three sides according to their relation with the angle. The little box in the right corner of the triangle given below denotes the right angle which is equal to 90^o.

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

The side opposite to the right angle is the longest side of the triangle which is known as the hypotenuse (H).
The side that is opposite to the angle θ is known as the opposite (O).
The side which lies next to the angle is known as the Adjacent(A)

Pythagoras theorem states that,

In any right-angled triangle,

[{left( {{mathbf{Opposite}}} right)^{mathbf{2}}} + {left( {{mathbf{Adjacent}}} right)^{mathbf{2}}} = {left( {{mathbf{Hypotenuse}}} right)^{mathbf{2}}}]

Trigonometric Ratios for Sec 30-

The six trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant of any angle. To calculate the trigonometric ratios of 30°, the knowledge of trigonometric ratios of standard angles and half angles is mandatory. Also, one should have knowledge of a few important trigonometric formulas.

As our angle of interest is sec 30, so accordingly sec of an angle is the ratio of the length of the longest side that is known as hypotenuse to the adjacent side of the angle. Secant is the reciprocal of cosine. To find the sec 30 degrees value, knowing the sine and cosine values of standard angles are important.

Sec 30 Formula –

[Sec{text{ }}30{text{ }}Value{text{ }} = {text{ }}frac{{Hypotenuse}}{{Adjacent}}]

Derivation of Sec 30 value –

Let’s consider a right-angled triangle; the secant of [angle alpha ] in the triangle below is a ratio of the length of the hypotenuse (longest side) and the adjacent side to the angle.

Here, angle [angle alpha ] is the angle formed between the adjacent side and the hypotenuse.

We know that, secant [angle alpha {text{ }} = {text{ }}frac{{Hypotenuse}}{{Adjacent{text{ }}Side}}{text{ }} = {text{ }}frac{{Hypotenuse}}{{Base}}]

Secant[angle alpha = frac{h}{b}]

From the reciprocal relation we know that, sec [angle alpha ] can be written as = [frac{1}{{cos angle alpha }}]

[sec{text{ }}{30^o}; = frac{1}{{cos{text{ }}{{30}^o}}}.]

Therefore, the value of sec [{30^o} = frac{2}{{sqrt 3 }}]

In the same way, we can write values of the important functions of Sin can also be determined by the giv
en method:

[Sin{text{ }}0^circ {text{ }} = {text{ }}sqrt {left( {frac{0}{4}} right)} ]
[;Sin{text{ }}30^circ {text{ }} = {text{ }}sqrt {left( {frac{1}{4}} right)} ]
[;Sin{text{ }}45^circ {text{ }} = {text{ }}sqrt {left( {frac{2}{4}} right)} ]
[;Sin{text{ }}60^circ {text{ }} = {text{ }}sqrt {left( {frac{3}{4}} right)} ]
[;Sin{text{ }}90^circ {text{ }} = {text{ }}sqrt {left( {frac{4}{4}} right)} ]

Simplifying in a Tabular Form:

	0°	30°	45°	60°	90°
Sin	0	[frac{1}{2}]	[frac{1}{{sqrt 2 }}]	[frac{{sqrt 3 }}{2}]	1

The value of cosine functions is opposite if sine functions as in:

[;Cos{text{ }}0^circ {text{ }} = {text{ }}Sin{text{ }}90^circ {text{ }} = 1]
[;Cos{text{ }}30^circ {text{ }} = {text{ }}Sin{text{ }}60^circ = frac{{sqrt 3 }}{2}]
[;Cos{text{ }}45^circ {text{ }} = {text{ }}sin{text{ }}45^circ = frac{1}{{sqrt 2 }}]
[;Cos{text{ }}60^circ {text{ }} = {text{ }}sin{text{ }}30^circ = {text{ }}frac{1}{2}]
[Cos{text{ }}90^circ {text{ }} = {text{ }}sin{text{ }}0^circ = 0]

The value of secant functions is the reciprocal of cosine functions as in:

[Sec;0^circ {text{ }} = {text{ }}frac{1}{{{text{ }}Cos{text{ }}0^circ }} = 1]
[Sec{text{ }}30^circ = {text{ }}frac{1}{{Cos{text{ }}30^circ }} = frac{2}{{sqrt 3 }}]
[;Sec{text{ }}45^circ {text{ }} = frac{1}{{Cos{text{ }}45^circ }} = sqrt 2 ]
[;Sec{text{ }}60^circ = frac{1}{{Cos{text{ }}60^circ }} = 2]
[;;Sec{text{ }}90^circ {text{ }} = {text{ }}frac{1}{{Cos{text{ }}90^circ }} = ;infty ]

Summary Table of the Value of Sin, Cos, Sec, Tan and Cosec Angles:

Angle	0°	30°	45°	60°	90°
Sin	0	[frac{1}{2}]	[frac{1}{{sqrt 2 }}]	[frac{{sqrt 3 }}{2}]	1
Cos	1	[frac{{sqrt 3 }}{2}]	[frac{1}{{sqrt 2 }}]	[frac{1}{2}]	0
Tan	0	[frac{1}{{sqrt 3 }}]	1	[sqrt 3 ]	[;{text{ }}infty ]
Cot	∞	[sqrt 3 ]	1	[frac{1}{{sqrt 3 }}]	0
Sec	1	[frac{2}{{sqrt 3 }}]	[sqrt 2 ]	2	∞
Cosec	∞	2	[sqrt 2 ]	[frac{2}{{sqrt 3 }}]	1

Questions to be Solved on Sec 30 Degree-

Question 1) Compute the value of the given question:

[;2sec30{;^o} + {text{ }}2cos{60^o}]

Solution) The given information is, 2 sec 30o + 2 cos 60o

We know that the value of sec 30o = [frac{2}{{sqrt 3 }}] and the value of cos 60o= [frac{1}{2}]

Now let’s substitute the values of the following,

[ = 2 times frac{2}{{sqrt 3 }} + 2 times frac{1}{2}]

[ = frac{4}{{sqrt 3 }} + 1]

Therefore, the value of [2sec{30^0} + 2cos{60^0}is{text{ }}equal{text{ }}to;4 + frac{{sqrt 3 }}{3}.]

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