Unit-6.2 | Trigonometric Ratios of Standard Angles and Any Angles | Class-09


What are standard angles?
Trigonometric values of angles such as 0°, 30°, 45°, 60°, and 90°, etc. that can be derived geometrically are known as standard angles.


Values of Trigonometric Ratios of Standard Angles:
Understanding the values of Trigonometric Ratios is essential. There are formulas that will allow you to find the value of all six functions i.e. sin, cos, tan, cosec, sec, and cot at various degrees.

We will need to understand the values of Trigonometric ratios at some specific angles ("standard angles") i.e. 0°, 30°, 45°, 60°, and 90°.

Here are the values of all trigonometric ratios of standard angles:

Values at 0°
sin 0°1
cos 0°0
tan 0°0
cosec 0°1
sec 0°0
cot 0°undefined

Values at 30°
sin 30°$\frac{1}{2}$
cos 30°$\frac{\sqrt{3}}{2}$
tan 30°$\sqrt{3}$
cosec 30°2
sec 30°$\frac{2}{\sqrt{3}}$
cot 30°$\frac{1}{\sqrt{3}}$

Values at 45°
sin 45°$\frac{1}{\sqrt{2}}$
cos 45°$\frac{1}{\sqrt{2}}$
tan 45°1
cosec 45°$\sqrt{2}$
sec 45°$\sqrt{2}$
cot 45°1

Values at 60°
sin 60°$\frac{\sqrt{3}}{2}$
cos 60°$\frac{1}{2}$
tan 60°$\frac{1}{\sqrt{3}}$
cosec 60°$\frac{2}{\sqrt{3}}$
sec 60°2
cot 60°$\sqrt{3}$

Values at 90°
sin 90°0
cos 90°1
tan 90°undefined
cosec 90°0
sec 90°1
cot 90°0


The basic function, we will read is CAST formula. It contains four different quadrant and represents formulas in a graph that can be easily understood and memorized.

But before this we will need to understand the values of Trigonometric functions at some specific ratios i.e. 0, 30, 45, 60, and 90.

For easy understanding, right now we will learn the values of sin, cos and tan only and we will learn others step by step.
 
Now that we know the values of these three functions at five different ratios, how can we understand them all? Okay, let us do the following:

$0° = \sqrt{\dfrac{0}{4}}$
$30° = \sqrt{\dfrac{1}{4}}$
$45° = \sqrt{\dfrac{2}{4}}$
$60° = \sqrt{\dfrac{3}{4}}$
$90° = \sqrt{\dfrac{4}{4}}$

So, sin0° = 0, sin30° = 1/2, sin45° = 1/√2, sin60° = √3/2, and sin90° = 1.

This way we can know the value of sin at the five different ratios.

Now what about cos? It's so simple. Just write the value of sin inversely and you will get the value of cos. 

Which means: cos0° = 1, cos30° = √3/2, cos45° = 1/√2, cos60° = 1/2, and cos90° = 0.

Sin and Cos are now easy. Tan??? I guess you might have already figured out the idea for tan! If not, I am here to guide you all.

Divide the values of sin and cos at the different ratios and simply you can get the value of tan.

That we know the values of sin, cos and tan where 'theta' is greater than or equal to 0 and lesser than or equal to 90, let us discuss the CAST FORMULA.

This is the general structure of the CAST FORMULA. Let's understand it's Trigonometric meaning.


It starts getting a bit more vast. So, remember every step from now on. 'C' in CAST stands for  'cos and sec', 'A' stands for 'All i.e. cos, sec, sin, cosec, tan and cot', 'S' stands for 'sin and cosec' and 'T' stands for ' tan and cot'.

Let's move further in detail.

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We have now added the values. Let me make you all more clear in another diagram. 

Remember 

A is in 1st quadrant.
S is in 2nd quadrant.
T is in 3rd quadrant.
C is in 4th quadrant.

Now,

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Remember these relationships.
I have no specific idea to make you more clear about this but if you have any comments make sure to leave it in the comment section but a little thing might help you if you are confused about them.

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Finally some more information and then we will be done with the CAST FORMULA 👍


  • For 0, 180 and 360 , all ratios are unchanged.
  • For 90 and 270 all ratios are changed as:. Cos=sin.   Cosec=Sec.      And Tan=Cot.
Lastly one more information about CAST FORMULA

signs (i.e. positive or negative) are assigned according to the cast formula.

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Now I am giving you the Trigonometric Ratios we can obtain from the CAST FORMULA

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Now you should be able to understand these all formulas that can be understood from Cast formula. So this is much for today's blog.

We will meet soon with Trigonometric questions and answers.


#CASTFORMULA
#COMPOUNDANGLES
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