Sin Theta Formula

Trigonometry, a branch of mathematics, is a powerful tool that helps us understand the relationships between the angles and sides of triangles. One of the fundamental concepts in trigonometry is the sine function, often represented as sin(θ), where θ is an angle.

This article will delve into the sin theta formula and its applications.

Understanding Sin Theta

In trigonometry, we often use the Greek letter θ (theta) to represent an angle. This angle can be in a triangle, a circle, or any geometric shape where the concept of an angle is applicable.

Common trigonometric values for sine (sin) of angles in degrees:

Sin θ Degree	Value
0°	0
30°	½
45°	1√2
60°	3√2
90°	1
180°	0

Trigonometric Ratios

Trigonometric ratios are ratios of the lengths of the right-angled triangle. These ratios can be used to determine the ratios of any two sides out of a total of three sides of a right-angled triangle.

• Sine Function: Sine ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its opposite side to its hypotenuse. i.e.,

Sin θ = AB/AC

• Cosine Function: Cosine ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its adjacent side to its hypotenuse. i.e.,

Cos θ = BC/AC

• Tangent Function: Tangent ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its opposite side to its adjacent. i.e.,

Tan θ = AB/BC

• Cotangent Function: Cotangent ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its adjacent side to its opposite. It’s the reciprocal of the tan ratio. i.e.,

Cot θ = BC/AB = 1/Tanθ

• Secant Function: Secant ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its Hypotenuse side to its adjacent. i.e.,

Sec θ = AC/BC

• Cosecant Function: Cosecant ratio for the given angle θ in a right-angled triangle is defined as the ratio of lengths of its hypotenuse side to its opposite. i.e.,

Cosec θ = AC/AB

Sin Theta Formula

In a Right-angled triangle, the sine function or sine theta is defined as the ratio of the opposite side to the hypotenuse of the triangle. In a triangle, the Sine rule helps to relate the sides and angles of the triangle with its circumradius(R) i.e, a/SinA = b/SinB = c/SinC = 2R. Where a, b, and c are lengths of the triangle, and A, B, C are angles, and R is circumradius.  

Sin θ = (Opposite Side / Hypotenuse)

From the above figure, sine θ can be written as 

sin θ = AB / AC

According to the Pythagoras Theorem,

We know that  AB² + BC² = AC²

Dividing both sides by AC²

⇒ (AB/AC)² + (BC/AC)² = (AC/AC)²

⇒ Sin²θ + Cos²θ = 1

Formula of Sin Theta

Various formulas related to sin θ are,

• Sin (- θ) = – sin θ
• Sin (90 – θ) = cos θ
• Sin (180 – θ) = sin θ
• sin² θ + cos² θ = 1
• sin (A+B) = Sin A×Cos B + Cos A×Sin B
• sin (A-B) = Sin A×Cos B – Cos A×Sin B
• Sin 2 θ = 2 sin θ. cos θ
• Sin 3 θ = 3 sin θ – 4 sin³ θ

Applications of Sin Theta Formula

Sin theta formula has a wide range of applications, from physics to engineering to computer graphics. It is used to model periodic phenomena such as sound and light waves, to calculate distances in navigation and geography, and to rotate and transform objects in computer graphics, among other things.

Conclusion

Sin theta formula is a cornerstone of trigonometry and a vital tool in various fields of study and industry. Understanding this formula and its applications can open doors to understanding more complex mathematical concepts and solving real-world problems.

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Sample Problems of Sin Theta Formula

Problems 1: If the sides of the right-angled triangle △ABC which is right-angled at B are 7, 25, and 24 respectively. Then find the value of SinC?

Solution:

Given:

As we know that Sin θ = (Opposite side/hypotenuse) 

Sin C = 24/25

Problems 2: If two sides of a right-angled triangle are 3 and 5 then find the sine of the smallest angle of the triangle?

Solution:

By Pythagoras theorem, other side of the triangle is found to be 4.

As the smaller side lies opposite to the smaller angle, 

Then Sine of smaller angle is equal to 3/5.

Problems 3: If sinA = 12/13 in the triangle △ABC, then find the least possible lengths of sides of the triangle?

Solution:

As we know, Sinθ = opposite/hypotenuse

Here, opposite side = 12 and hypotenuse = 13

Then by the pythagoras theorem, other side of the triangle is 5 units

Problems 4: If the lengths of sides of a right-angled △PQR are in A.P. then find the sine values of the smaller angles?

Solution: 

Only possible  Pythagorean triplet for the given condition is (3, 4, 5).

Therefore, the sine values of the smaller sides are 3/5 and 4/5

Problems 5: In a triangle △XYZ if CosX=1/2 then find the value of SinY?

Solution:

From given data, angle X is equal to 60 degrees, then Y=30 degrees as it’s a right angled triangle.

Therefore, Sin Y = Sin30°

Y = 1/2

Problems 6: If sinθ.Secθ = 1/5 then find the value of Sinθ?

Solution:

As secθ = 1/cosθ 

Secθ = Tanθ = 1/5.

Therefore, opposite side= k and adjacent side is 5k and hypotenuse = √26 k.

Then Sinθ = k/√26 k

= 1/√26

Problems 7: In a right-angled triangle, if the ratio of smaller angles is 1:2 then find the sum of sines of smaller angles of the triangle?

Solution:

Let the smaller angles be A, B. As A:B = 1:2.

So, A = k and B = 2k. As A + B = 90. 

⇒ k + 2k = 90

⇒ k = 30.

Therefore, the other angles are 30 and 60

So, their sine values are 1/2 and √3/2

Therefore, the sum of the sines is (1 + √3)/2

FAQs on Sin Theta Formula

What does θ represent in the sin theta formula?

In the sin theta formula, θ (theta) represents an angle. This angle can be in a triangle, a circle, or any geometric shape where the concept of an angle is applicable.

Where is the sin theta formula used?

The sin theta formula has a wide range of applications, including modeling periodic phenomena such as sound and light waves, calculating distances in navigation and geography, and rotating and transforming objects in computer graphics.

Are there other trigonometric functions similar to sine?

Yes, there are other trigonometric functions similar to sine, including cosine (cos) and tangent (tan). These functions also relate the angles and sides of a triangle.

Is Sine continuous?

Yes, sine function is a continuous function as the graph of sine function has no breaks and gaps.