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Trigonometry is the branch of mathematics that deals with the relationship between angles and length (not arc length) of a line made by angle. Whenever we talk about angles, we need to have 2 intersecting lines, through which angle has been made. Adding to that we need one more line on which we’ll measure the length. So, in a nutshell, we need three lines. That’s what “tri” stands for. Now there are many types of triangles we know. Which type of triangle should be? So, the triangle should be right-angled. Now we’ll see different types of trigonometric ratios.

We know in trigonometry we are going to discuss the relationship between angle and length in right angled triangles. Let’s first remind ourselves, what we call sides of the right angled triangle.

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Observe that above the right angle triangle ABC is right angled at C. If we consider angle∠B then side AC will by perpendicular, side BC will be called based and side AB is called hypotenuse. Similarly, If we consider angle ∠A then BC will be perpendicular, AC will be considered base and AB will hypotenuse. Among these three, different ratios will give sin, cos, tan, cot, sec, cosec ratios.

So now we’ll discuss what sin theta and sin formula are. As we have discussed above, sin is a trigonometric ratio that is perpendicular/hypotenuse.

sin x = \[\frac{perpendicular}{Hypotenuse}\]

Here x represents the angle which is under consideration.

1. Cos x = \[\frac{3}{5}\] Then Find the Value of Sin x.

Solution: We know that, cos θ = \[\frac{Base}{Hypotenuse}\]

On comparing the given ratio, Base = 3, Hypotenuse= 5.

Now we also know Pythagoras theorem, which says,

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

⇒ (Perpendicular)² = (Hypotenuse)² - (Base)²

⇒ (Perpendicular)² = 5² - 3²

⇒ (Perpendicular)² = 25 - 9

⇒ (Perpendicular)² = 16

⇒ (Perpendicular)² = 4

Here we are considering only positive signs because the length of the side can’t be negative.

Sin x = \[\frac{4}{5}\]

Hence this is the required answer.

2. If Cosec x = \[\frac{6}{7}\] Then Find the Value of Sin x.

**Solution:** We know that sin x = \[\frac{1}{\text{cosec x}}\]

Using the above value we have, sin x =\[\frac{1}{\frac{6}{7}}\]

Hence, sin x = \[\frac{7}{6}\]

FAQ (Frequently Asked Questions)

1. Question: Who is the Founder of Trigonometry?

Answer: A greek mathematician, Hipparchus discovered the concept of trigonometry. He has shown his tremendous work in geography and astronomy.

2. Question: What are the Primary Trigonometric Functions?

Answer: The Sin function, cosine function and the tangent function are three primary trigonometric functions.

3. What is the Formula for Six Trigonometric Functions?

Answer: The formula for six trigonometric functions are:

Sin x = Perpendicular /Hypotenuse

Cos x = Base / Hypotenuse

Tan x = Perpendicular / Base

Cot x = Base / Perpendicular

Sec x = Hypotenuse / Base

Cosec x = Hypotenuse / Perpendicular

4. What is Sin Theta?

Answer: sin theta is a ratio of perpendicular and hypotenuse. In maths sin theta or sin function ranges it’s values from -1 to 1. It’s domain is whole real numbers.

5. What is the Value of x in 1 Sin x Formula?

Answer: When 1 sin theta is equal to 1 then we say that x is equal to π/2. You can see the standard values of sinx formula in following table

| 0⁰ | 30⁰ | 45⁰ | 60⁰ | 90⁰ |

| 0 | ½ | 1/√2 | √3/2 | 1 |