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.Â
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. Â
Sin θ = (opposite side / hypotenuse)
From the above figure, sine θ can be written asÂ
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)Â
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,Â
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:Â
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/cosθÂ
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.Â
⇒ 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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