Trigonometry

Trigonometry is a branch of mathematics in which we study the relationship between the angles and sides of triangles, mainly right-angled triangles. It helps us solve problems with lengths, angles, and heights. 

From designing bridges and launching rockets to finding our way home using GPS, trigonometry is everywhere. 

Hey, did you know that triangles are important to solve some of the trickiest problems? Trigonometry is an interesting topic that helps you learn about the relationship between the angles and sides of the triangles, mainly right-angled triangles. It plays a critical role in many fields such as physics, engineering, and everyday situations like navigation and architecture. 

Have you ever wondered who thought of using triangles to solve real-world problems? Let's make this a fun trip to look back and know the history of trigonometry- because math did not always come from textbooks.

 

Where it all started…

 

People started noticing that triangles could help them measure things, 

 

 

• Ancient Egyptians used the early forms of trigonometry to build pyramids.

 

• The Greeks shared an equal interest in triangles. Hipparchus, the father of trigonometry, created the first trigonometric tables.

 

• Indian mathematicians, like Aryabhata, came up with advanced ideas about angles.

 

• Further, the Islamic scholars came up with more ideas, like using trigonometry to navigate across deserts and seas.
   

In trigonometry, the most important triangle is the right angles triangle. This triangle always has:

 

• One right angle 90°

 

• Two smaller angles which are less than 90°

 

• Right-angled triangle: A triangle with one angle exactly 90 degrees, like the corner of a room.

 

• Let’s understand the three sides of the right-angled triangle:

 

 

• Hypotenuse: it is the longest side/arm of the triangle.  

 

• Opposite side: The side that is facing directly to the angle you are supposed to solve.

 

• Adjacent side: The side next to the angle you are focusing on, but it is not the hypotenuse.

 

• Pythagoras theorem: This is a very important rule that helps us solve a triangle.It is given as:

 

(Hypotenuse)²=(Opposite side)² + (Adjacent side)²

 

 

Alright, kids! Imagine you are on a treasure hunt, and the treasure is hidden at the top of a tree. Know the distance from where you are standing and the angle between the ground and your line of sight. But how will you find the height of the tree without climbing it?

 

Oh, here's when trigonometry comes to the rescue!

 

Let's meet the three most important ratios:

 

Sine (sin)- The height finder 

 

Imagine standing on the ground and looking up at the treasure on top of the tree. 

The sine ratio is compared to the height of the tree, which is opposite to the ladder, i.e., the hypotenuse.  Sin θ=opposite / hypotenuse.

 

You can remember it as SOH for sin.

Cosine (cos) - the adjacent-to the hypotenuse

.
Now, if you are measuring how far you are from the tree trunk, you are dealing 
with the adjacent side.

Cos θ =Adjacent / hypotenuse.

 

You can remember it as CAH for cos.

 

Tangent (tan)- The angle decider

If you want to compare the height of the tree to how far away you are standing,

you use the tangent ratio.

Tan  θ=opposite / adjacent 

Tan θ = sin θ /cos θ 

You can remember it as TOA for tan. 

The trigonometric ratios are also known as trigonometric functions. Sine, Cosine, and Tangent are the three important functions in trigonometry, the other functions are the inverse of the three functions. 

 

Cosecant (COSEC)- The reverse of sine 

It is the reciprocal of sine, we flip the sine fraction. It is how we compare the hypotenuse to the opposite side.

 

  COSEC θ=hypotenuse / opposite. COSEC  θ= =1 /sinθ

 

Secant (sec)- The reverse of cosine

Secant is the reciprocal of cosine, we flip the cosine fraction. It is how we compare the hypotenuse to the adjacent side.

 Sec θ=hypotenuse / Adjacent. Sec θ= 1/cosθ

 

Cotangent (cot)- The reverse of tangent

Cotangent is the reciprocal of tangent. It is how we compare the adjacent side to the opposite side.

 Cot θ=Adjacent / opposite. Cot θ= 1/ tanθ

 

Odd functions: The functions change signs when the angle is negative.

 

sin(-x)= -sin(x)

tan(-x)= -tan(x)

COSEC(-x)= -COSEC(x)

 

Even functions: The functions stay the same when the angle is negative.

Cos(-x)= cos(x)

sec(-x)= sec(x)
 

Trigonometric ratios can be applied to common angles. The table given below shows the value of trigonometric ratios at certain values

. 

Lets’s learn about how to create a trigonometric ratios table.
 

 

Step 1: write the first five whole numbers 

Let’s start by writing the first five whole numbers. Imagine we are counting like a game of hopscotch, with some spaces in between each number so we can move to the next position.

0 | 1 | 2 | 3 | 4

 

Step 2: Divide each number by 4

Now, we divide each number by 4. Think you are sharing cookies with your friends, each friend gets a piece. 

0 | 1/4 |  2/4 |  3/4 |  4/4 

When we simplify 2/4  and 4/4 , so we write it as:

0 | 1/4 |  1/2 |  3/4 |  1 

 

Step 3: Take the square root of each resulting number

0 | √1/4 | √1/2 |  √3/4 | √1 

Simplifying this we get,

Step 4: Sine values for angles 0°,30°, 45°, 60° and 90°

 

Sine : 0 | 1/2 | 1/√2 | √ 3/2| 1 

 

Step 5: Reverse the order for cosine values

Now, let's have some fun and reverse the order of sine values to find the cosine values. 

1 | √3/2 | 1/√2 |1/2 | 0

 

Step 6: Find the Tangent values

 Now we find the tangent values, which are simply the sine that is divided by cosine.

Tan 0 = 0/1 =0

Tan 30= 1/2 / √3/2 =1/√3

 

Tan 45= 1/√2 / 1/√2 = 1

 

Tan 60= √3/2 / 1/2 =√3

 

Tan 90 = 1/0 = Not defined.

 

Trigonometric formulas are the rules that help us understand the relationship between angles and sides in triangles, specifically right-angled triangles. By using the formulas, we can also solve many real-life problems.

 

1.Pythagoras trigonometric identities 

Sin² θ + cos²θ  =1

1+tan²θ=sec²θ

COSEC² θ = 1+ cot² θ

 

2.Double angle identities

Sine: sin2θ= 2sinθcosθ

Cosine: cos2θ= cos²θ-sin²θ

Tangent: tan2θ=(2tanθ)/(1-tan²θ)

 

3.Product - Sum identities

SinA+sinB=2sin(A+B/2)cos (A-B/2)

SinA-sinB=2cos(A+B/2)sin (A-B/2)

CosA+cosB=2cos(A+B/2)cos (A-B/2)

CosA-cosB= - 2sin(A+B/2)sin (A-B/2)

4.Sum and difference identities 

sin(A+B)= sinAcosB + cosAsinB

sin(A-B)= sinAcosB - cosAsinB

cos(A+B)= cosAcosB - sinAsinB

cos(A-B)= cosAcosB + sinAsinB

tan(A+B)=(tanA+tanB)/(1-tanAtanB)

               tan(A-B)=(tanA-tanB)/(1+tanAtanB)
 

1.SOH-CAH-TOA helps you remember the basics of sine, cosine, and tangent ratios:

 

SOH: sine= opposite/hypotenuse

CAH: cosine= adjacent/hypotenuse

TOA: tangent= opposite/adjacent

 

2.You can also create a fun sentence/phrase to memorize, “Silly Owls Help Cats And Turtles”.

 

3.You can also make a trigonometric ratio triangle.  Draw a triangle and label them as opposite, adjacent, and hypotenuse.

 

4.Use color coding for different ratios and functions.