Sec 30

What is Trigonometry?

  • In Mathematics, trigonometry is a branch that deals with triangles. 

  • The word Trigonometry is composed of two Greek words trigōnon (meaning triangle) and metron (meaning measure). 

  • So in short, we can say that measuring a triangle (specifically right-angled triangle) is trigonometry.

  • Trigonometry is the study between the relationships dealing with angles, heights, and lengths of triangles, and also the relationships between the different circle parts and other geometric figures. 

  • In the field of astronomy, engineering, architectural design, and physics, trigonometry applications are found. There are a huge number of fields in which trigonometry identities and trigonometric equations are used.

 

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 \[\frac{1}{{Cosec\left( x \right)}}\]

\[cosec{\text{ }}\left( x \right)\] can be written as \[\frac{1}{{Sin\left( x \right)}}\]

\[\cos \left( x \right)\]can be written as \[\frac{1}{{sec\left( x \right)}}\]

\[sec\left( x \right)\] can be written as \[\frac{1}{{Cos\left( x \right)}}\]

\[tan\left( x \right)\] can be written as \[\frac{1}{{cot\left( x \right)}}\]

\[cot{\text{ }}\left( x \right)\] can be written as \[\frac{1}{{tan\left( x \right)}}\]

 

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 90o.

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 given 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:


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 \[2sec{30^0} + 2cos{60^0}is{\text{ }}equal{\text{ }}to\;4 + \frac{{\sqrt 3 }}{3}.\]

FAQ (Frequently Asked Questions)

Question 1) What is Pythagoras Theorem?

Answer) Pythagoras theorem is applicable only on a right-angled triangle. It explains that in a right-angle triangle, the square of the hypotenuse will always be equal to the sum of the squares of the opposite side and the square of the adjacent side.


Where, c = hypotenuse, a = base and b = perpendicular. 


Pythagoras Theorem formula

(Hypotenuse)2 = (Base)2+ (Perpendicular)2

Question 2) What is Tan 30 as a Fraction?

Answer) The value of tan 30° in fraction is equal to,

Since, we know that tan 30 degrees = sin 30°/cos 30° = 1/2/√3/2 = 1/√3.

Question 3) What is the Exact Value of Sec 30 Degrees?

Answer) The exact value of secant 30 degrees is equal to 2/√3.