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Sine formula is used in the branch of trigonometry to find the angle of a right-angled triangle.
- This formula showcases the relation between the sides and the angle of a right-angled triangle.
- Sine is the ratio between the hypotenuse and the opposite side of the hypotenuse.
- The sine formula is one of the functions of Trigonometry and is used in other branches such as calculus, logarithms, etc.
Read More: Trigonometry Table
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Key Terms: Trigonometry, Triangle, Angles, Ratio, Laws of sine, Length, Area, Sine.
Sine Formula
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Sine Formula helps us find the sides or the angles of an oblique triangle. If all its sides are given, we can find any of the angles while if two sides and the angle between the two is given, we can find the third side’s length.
- The ratio of an angle's opposing side to its hypotenuse is known as the sine function.
- Other names for the Law of Sines are Sine Law, Sine Rule, and Sine Formula.
- According to the "Sine Rule," a triangle's side lengths and the sine of each of its opposing angles must be equal.
- For all three sides and opposing angles, the ratio is the same.
Read More: Sin Cos Formulas
Law of Sines
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Sine rule can be used in two cases:
- If two angles and one side in the triangle are given.
- If two sides and a non-included angle are given.
The rule says that the sines of the opposite angles of a triangle are proportional to the sides of a triangle.
a/sinA = b/sinB = c/sinC
Read More: Sin 30 Degrees
Derivation of Sine Formula
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For the derivation of the sine formula, let us draw an altitude through B and term it hb. Now to express hb in terms of side and sine of the angle:
Sin A = hb/c
hb = c sin A
sin C = hb/a
hb = a sin C
Equating the two hb
hb= hb
c sin A= a sin C
c/sin C = a/sin A
Now, to include a third angle B and side b in the above, construct an altitude as hc
Sin A = hc/b
hc = a sin B
hc = hc
b sin A = a sin B
b/sin B = a/sin A
a/sin A = b/ sin B = C/ sin C
Sine function is the ratio of the side of the triangle opposite to angle and divided by the hypotenuse.
Read More: Law of Tangents
Applications of Sine Formula
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The law helps find the missing side or angle of a triangle if the data is given, but the law can also be applied to calculate the following:
- The length of the side of a triangle using ASA or AAS criterion.
- The unknown angle of a triangle.
- The area of a triangle.
Read More: Sine Squared X Formula
Things to Remember
- Sin function is used in the branch of Trigonometry.
- Sin function is the ratio between the height and hypotenuse of a right-angled triangle.
- The ratio between the hypotenuse and the other side of the hypotenuse is known as sine.
- The Law of Sines is sometimes referred to as the Sine Law, Sine Rule, and Sine Formula.
- Sines of opposite angles of a triangle are proportional to the sides of a triangle
- Sin rule can be used if two angles and one side are given or two sides and a different angle are mentioned.
Sample Questions
Ques: Calculate the sine angle of a right triangle whose opposite side and hypotenuse are 10 cm and 12 cm respectively. (2 Marks)
Ans: Given details:-
Opposite side = 10 cm
Hypotenuse = 12 cm
Sin theta = opposite/hypotenuse
= 10/12
= 0.83
Ques: If sin A = 0.5, find the value of X. (3 Marks)
Ans: Given, Sin A = 0.5
Sin theta = opposite/Hypotenuse
BC/AC = ½
12/AC = ½
AC = 12 x 2 = 24 cm
By Pythagoras theorem, AC2 = AB2 + BC2
242 = x2 + 1212
X2 = 576 – 144
X2 = √452
X = 20.78 cm
Ques: Solve triangle PQR in which angle P = 63.5 and angle Q = 51.2 and r = 6.3 cm. (3 Marks)
Ans: We know the two angles so we can find out the third angle.
Angle R = 180 - (63.5 + 51.2) = 65.3
Now to the sides,
a/sin A = b/ sin B = c/ sin C
6.3/sin 63.5 = p/ sin 63.5
P = 6.21 cm
Similarly q = 5.4 cm
Thus Angle R = 65.3 degrees.
Ques: If angle B = 21°, angle C = 46°¦ and the side AB = 9 cm. Find other sides of the triangle. (3 Marks)
Ans: Two angles and a side are given.
The sum of angles of a triangle is 180°¦
Angle A = 113°
AB = C = 9 cm
Using the sine rule:
a/sin 113°¦ = b/ sin21° = 9/sin46°
b= sin 21° x 9/sin 46°
= 4.484 cm
A = sin 113° x 9/sin 46°
= 11.517 cm
Ques: Given a = 20 units c = 25 units and Angle C = 42°. Find angle A of the triangle. (3 Marks)
Ans: By using sine law:
20/sin A = 25/sin 42°
Sin A/20 = sin 42°/25
Sin A = (sin 42°/25) x 20
Sin A = (0.6691/5) x 4
Sin A = 0.5363
A = sin-1 (0.5363)
A= 32.36°
Ques: If angle B = 21° angle c = 46° and side AB = 9 cm then solve the triangle. (3 Marks)
Ans: Given are two angles and a side.
Using the law of sines, the sum of the angles of a triangle is 180°
Therefore, angle A = 113°
AB = C = 9 cm.
a/sin113° = b/sin21° = 9/sin46°
b/sin21° = 9/sin 46°
b= sin 21°x 9/sin46°
= 4.484 cm
A sin 113° x 9/sin46°
= 11.517 cm
Ques: In PQR, Sin P = 1/3 and Sin Q = ¼. Find the ratio of side p/side q. (2 Marks)
Ans: Using the law of sine,
p/Sin P = q/sin Q = p/1/3 = 1/1/4
1/3p = 1/4q
4/1 x 1/3q = p
4/3 = p/q
Therefore, the ratio of the side p to the side q is 4:3
Ques: Find angle P and angle Q and the length of the third side when angle R = 36° and angle p = 2.5 cm and r = 7 cm. (3 Marks)
Ans: From the sin rule p/sin P = q/sin Q = r/sin R
2.5/sin P = q/sin Q = 7/sin 36°
2.5/sin P = 7/sin 36° [sin 36° = 0.5857]
Sin P = 0.20992
P = sin-1 (0.20992)
Angle P = 12.12°
Angle P + Angle Q + Angle R = 180°
12.12° + Angle Q + 36° = 180°
Angle Q = 131.88°
Q/sin 131.88° = 7/sin 36°
Q/0.7445 = 7/0.5878
q= 8.866 cm.
Hence, Angle P = 12.12°, Angle Q = 131.88° and q = 8.866cm.
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