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

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.