Sin Theta Formula Explained

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Trigonometry is derived from the Greek words trigonon and metron, which mean triangle and measure, respectively. Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant are the six trigonometry ratios. These trigonometry ratios describe the various combinations of a right-angled triangle.

 

Trigonometric ratios

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




Sine function: The 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., Sinθ = AB/AC

Cosine function: The 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, Cosθ = BC/AC

Tangent Function: The 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, Tanθ = AB/BC

Cotangent Function: The 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, Cotθ = BC/AB =1/Tanθ

Secant Function: The 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, Secθ = AC/BC

Cosecant Function: The 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, Cosecθ = 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 

Sinθ = AB / AC

According to the Pythagoras theorem, we know that  AB2 + BC2 = AC2. On dividing both sides by AC2

⇒ (AB/AC)2 + (BC/AC)2

⇒ Sin2θ + Cos2θ = 1

 

Sample Problems

Question 1: If the sides of the right-angled triangle â–³ABC which is right-angled at B are 7, 25, and 24 respectively. Then find the value of SinC?

Solution:




As we know that Sinθ = (Opposite side/hypotenuse) 

SinC = 24/25

 

Question 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.

 

Question 3: If sinA = 12/13 in the triangle â–³ABC, 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

 

Question 4: If the lengths of sides of a right-angled â–³PQR are in A.P. then find the sine values of the smaller angles?

Solution: 

The 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

 

Question 5: In a triangle â–³XYZ if CosX=1/2 then find the value of SinY?

Solution:

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

Therefore, SinY = Sin30°

Y = 1/2

 

Question 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

 

Question 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
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