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Sin Theta Formula

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The term Trigonometry is derived from Greek words i.e; trigonon and metron, which implies triangle and to measure respectively, θ. There are 6 Trigonometry ratios namely, Sine, Cosine, Tangent, Cotangent, Secant, and Cosecant. These trigonometry ratios tell the different combinations in a right-angled triangle. 

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: 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


Last Updated : 27 Mar, 2024
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