# Trigonometry is one of the most interesting chapters of Quantitative Aptitude section. Basically, it is a part of SSC syllabus. Today I will tell you the easy method to learn all the basics of trigonometry i.e. Trigonometric Ratios, facts and formulas.

Question
Trigonometry
Trigonometry is one of the most interesting chapters of Quantitative Aptitude section. Basically, it is a part of SSC syllabus. Today I will tell you the easy method to learn all the basics of trigonometry i.e. Trigonometric Ratios, facts and formulas.

2021-04-16

## Trigonometric Ratios

There are six trigonometric ratios. First three are the primary functions and last three are just the reciprocals of above three. Those are written as follows:

• $$(Sin \theta)$$
• $$(Cos \theta)$$
• $$(Tan \theta)$$
• $$(Cot \theta)$$
• $$(Sec \theta)$$
• $$(Cosec \theta)$$

See the following image, You are already well versed with this image. This is a basic right angled triangle.

Points to remember:
1. Usually, student got confused between perpendicular and base. Always remember that out of three sides, perpendicular is that side which is opposite to the given angle.

In following figure, perpendicular (P) is the side which is opposite to the purple shaded angle(given angle)

2. Side opposite to the right angle(shaded red)  is hypotenuse (H).

3. Remaining side is known as base (B).

Now, you must be knowing that all the trigonometric ratios can be derived from these sides. Let us see how.

$$Sin \theta = P/H$$                   $$Cosec \theta = {H/P}$$

$$Cos \theta = B/H$$                  ( $$Sec \theta = H/B$$

$$Tan \theta = P/B$$                       ($$Cot \theta = B/P$$

Learning all these formulae is somewhat difficult because trigonometry is full of formulas. Let me tell you the very easy method to learn the above formulas.

Following is the basic pattern which you should learn by heart. The easy way to remember this is the way our teachers teach us in schools. I am sure that is the best way to remember this. You can never forget this trick:

(Some People Have) (Curly Brown Hair) (Through Proper Brushing)

Learn the above line and write 3-3 words vertically like:

Some         Curly        Through
People       Brown       Proper
Have           Hair          Brushing

i.e.
S    C    T
P    B    P
H    H    B

 Sin Cos Tan P B P H H B Cosec Sec Cot

Now, It will seem very easy to learn the ratios. Isn't it?

## Basic Formula

Following are some very basic formulae. Go through these formulas once, then we will proceed to the trigonometric ratios of standard angle.

• $$(Cosec \theta = 1/ (Sin \theta)$$
• $$(Sec \theta = 1/ (Cos \theta)$$
• $$(Cot \theta = 1tan \theta)$$
• $$(Tan \theta = (Sin \theta)Cos \theta)$$
• $$(Sin^2\theta + Cos^2\theta = 1$$
• $$(1+ tan^2 \theta = sec^2 \theta)$$
• $$(1+ cot^2 \theta = cosec^2 \theta)$$

## Trigonometric Ratios of Standard Angles

You must have seen the following table of trigonometric ratios, but so many times you may get confused by remembering the values. Here, I will tell you the basic technique which will help you to learn this table.

 theta 0° 30° 45° 60° 90° Sin theta 0 1/2 $$(1/\sqrt2$$ $$(\sqrt3/2$$ 1 Cos theta 1 $$(\sqrt3/2$$ $$(1/\sqrt2$$ 1/2 0 Tan theta 0 $$(1/\sqrt3$$ 1 $$(\sqrt3$$ Not Defined Cot theta Not Defined $$(\sqrt3$$ 1 $$(1/\sqrt3$$ 0 Cosec theta 1 $$(2/\sqrt3$$ $$(\sqrt2$$ 2 Not Defined Sec theta Not Defined 2 $$(\sqrt2$$ $$(2/\sqrt2$$ 1

Firstly, to find the values of Row1

• Write numbers from 0 to 4
• Divide the numbers with 4
• Take the square root of all

($$0 \to 0/4\to 0$$                                                     0

($$1 \to 1/4 \to 1/4$$                                                     1/2

($$2 \to 2/4 \to 1/2$$      Taking Square root$$\to$$         $$1/\sqrt2$$

($$3 \to 3/4 \to 3/4$$                                            $$\sqrt3/2$$
($$4\to 4/4 \to 1$$                                                          1

See. how simple. You just need to divide 4 to numbers 0 to 4, then take square root.

Row2 = It is just the reverse of Sin values. (See table)

Row3 = Value of tan theta is derived by using ($$(Sin \theta) cos\theta$$)

Row4 = Use $$(Cot \theta = 1/ tan\theta)$$

Row5 = Use $$Sec (\theta= 1/ cos \theta)$$

Row6 = Use $$(Cosec \theta = 1/sin \theta)$$

## Examples with Solution

Example1: Find the value of:

$$((2tan30^o) / (1+ tan^2 30^o)$$

Solution:$$((2tan30^o) / (1+ tan^2 30^o) =(2/\sqrt3)1+(1/\sqrt3)^2)$$

Solving above will give = ($$\sqrt3$$

Example2: If  ($$tan\theta= \sqrt2/\sqrt3$$, then what will be the value of $$cos\theta$$?

Solution: ($$tan \theta = P/B$$

Therefore, ($$P = \sqrt2 and B = \sqrt3$$

using Pythagoras Theorem(, $$H^2 = P^2 + B^2$$

($$H^2 = 2+ 3 = 5$$

⇒ ($$H = \sqrt5$$

Therefore, ($$Cos \theta = B/H = \sqrt3/\sqrt5$$

I hope you understand this topic till now, I will soon update the rest of the topic.
Your valuable feedback will be appreciated.

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The table below shows the number of people for three different race groups who were shot by police that were either armed or unarmed. These values are very close to the exact numbers. They have been changed slightly for each student to get a unique problem.
Suspect was Armed:
Black - 543
White - 1176
Hispanic - 378
Total - 2097
Suspect was unarmed:
Black - 60
White - 67
Hispanic - 38
Total - 165
Total:
Black - 603
White - 1243
Hispanic - 416
Total - 2262
Give your answer as a decimal to at least three decimal places.
a) What percent are Black?
b) What percent are Unarmed?
c) In order for two variables to be Independent of each other, the P $$(A and B) = P(A) \cdot P(B) P(A and B) = P(A) \cdot P(B).$$
This just means that the percentage of times that both things happen equals the individual percentages multiplied together (Only if they are Independent of each other).
Therefore, if a person's race is independent of whether they were killed being unarmed then the percentage of black people that are killed while being unarmed should equal the percentage of blacks times the percentage of Unarmed. Let's check this. Multiply your answer to part a (percentage of blacks) by your answer to part b (percentage of unarmed).
Remember, the previous answer is only correct if the variables are Independent.
d) Now let's get the real percent that are Black and Unarmed by using the table?
If answer c is "significantly different" than answer d, then that means that there could be a different percentage of unarmed people being shot based on race. We will check this out later in the course.
Let's compare the percentage of unarmed shot for each race.
e) What percent are White and Unarmed?
f) What percent are Hispanic and Unarmed?
If you compare answers d, e and f it shows the highest percentage of unarmed people being shot is most likely white.
Why is that?
This is because there are more white people in the United States than any other race and therefore there are likely to be more white people in the table. Since there are more white people in the table, there most likely would be more white and unarmed people shot by police than any other race. This pulls the percentage of white and unarmed up. In addition, there most likely would be more white and armed shot by police. All the percentages for white people would be higher, because there are more white people. For example, the table contains very few Hispanic people, and the percentage of people in the table that were Hispanic and unarmed is the lowest percentage.
Think of it this way. If you went to a college that was 90% female and 10% male, then females would most likely have the highest percentage of A grades. They would also most likely have the highest percentage of B, C, D and F grades
The correct way to compare is "conditional probability". Conditional probability is getting the probability of something happening, given we are dealing with just the people in a particular group.
g) What percent of blacks shot and killed by police were unarmed?
h) What percent of whites shot and killed by police were unarmed?
i) What percent of Hispanics shot and killed by police were unarmed?
You can see by the answers to part g and h, that the percentage of blacks that were unarmed and killed by police is approximately twice that of whites that were unarmed and killed by police.
j) Why do you believe this is happening?
Do a search on the internet for reasons why blacks are more likely to be killed by police. Read a few articles on the topic. Write your response using the articles as references. Give the websites used in your response. Your answer should be several sentences long with at least one website listed. This part of this problem will be graded after the due date.