In the trigonometry, Sine rule formula (also called as sine formula or law of sines or sines law or sine rule) is the equation that can be used for relating the lengths of the sides of the arbitrary triangle to sines of the corresponding angle. According to the sine rule formula,
`a/sin A = b/sin B =c/sin C`
The sine rule formula in the trigonometry defines that the ratio of Sine of the angle with respect to its included sign and the vice versa depending on number of the properties in an obtuse triangle. If the obtuse triangle has two sides with given included angle as SSA(Side-Side-Angle) or ASA(Angle-Side-Angle) or AAS(Angle-Angle-Side). Then the Sine Rule Formula can be used to find the Angle of one of the sides.
a, b, and c are the lengths of sides of a given triangle, and A, B, and C are the opposite angles. Sometimes the sine rule formula is also stated as the reciprocal of this equation:
`sin A/a = sin B/b =sin C/c`
The sine rule formula is used to compute the remaining sides of the triangle when the two angles and one side is known. The technique is known as triangulation. It is used when two sides and one of the non-enclosed angle is known. In such cases, the sine rule formula gives two possible values for an enclosed angle.
Proof of the Sine rule formula
From the above triangle AXC,
`sin A=h/b`
`bsin A=h`
From triangle XBC,
`"sin B `
`asin B=h`
Equating both the Equations we have
`h=bsin A=asin B`
So,
`b/sin B=a/sin A`
A perpendicular from A to BC, we can show that
`b/sin B=c/sin C`
Hence we have the Sine Rule:
`a/sin A = b/sin B =c/sin C`
The area of a triangle :
The Area of any triangle is `1/2 ab sinC` using the sine rule formula.
The Sine rule formula can be used if we don't know the height of a triangle (since we have to know the height for
`1/2 (base) * (height)` .
Examples using the sine rule formula
Q 1: Given side a = 10, side c = 14, and angle C = 30°
Sol: Using the Sine rule formula , we conclude that
`sin A / 20 = sin 30 /14`
`sin A = 1/2 * 20/14`
`A = sin ^-1 (1/7)`
A = 0.14°
2) Applying the sine rule formula :
`17/sin 62 = 13/sin theta`
`17 sin theta = 13 sin 62`
`sin theta = (13 sin 62)/ 17`
`sin theta = (13 * 0.8829)/17`
`sin theta = 0.6752`
`theta = sin ^-1 (0.6752)`
? = 42.47°
`a/sin A = b/sin B =c/sin C`
The sine rule formula in the trigonometry defines that the ratio of Sine of the angle with respect to its included sign and the vice versa depending on number of the properties in an obtuse triangle. If the obtuse triangle has two sides with given included angle as SSA(Side-Side-Angle) or ASA(Angle-Side-Angle) or AAS(Angle-Angle-Side). Then the Sine Rule Formula can be used to find the Angle of one of the sides.
a, b, and c are the lengths of sides of a given triangle, and A, B, and C are the opposite angles. Sometimes the sine rule formula is also stated as the reciprocal of this equation:
`sin A/a = sin B/b =sin C/c`
The sine rule formula is used to compute the remaining sides of the triangle when the two angles and one side is known. The technique is known as triangulation. It is used when two sides and one of the non-enclosed angle is known. In such cases, the sine rule formula gives two possible values for an enclosed angle.
Proof of the Sine rule formula
From the above triangle AXC,
`sin A=h/b`
`bsin A=h`
From triangle XBC,
`"sin B `
`asin B=h`
Equating both the Equations we have
`h=bsin A=asin B`
So,
`b/sin B=a/sin A`
A perpendicular from A to BC, we can show that
`b/sin B=c/sin C`
Hence we have the Sine Rule:
`a/sin A = b/sin B =c/sin C`
The area of a triangle :
The Area of any triangle is `1/2 ab sinC` using the sine rule formula.
The Sine rule formula can be used if we don't know the height of a triangle (since we have to know the height for
`1/2 (base) * (height)` .
Examples using the sine rule formula
Q 1: Given side a = 10, side c = 14, and angle C = 30°
Sol: Using the Sine rule formula , we conclude that
`sin A / 20 = sin 30 /14`
`sin A = 1/2 * 20/14`
`A = sin ^-1 (1/7)`
A = 0.14°
2) Applying the sine rule formula :
`17/sin 62 = 13/sin theta`
`17 sin theta = 13 sin 62`
`sin theta = (13 sin 62)/ 17`
`sin theta = (13 * 0.8829)/17`
`sin theta = 0.6752`
`theta = sin ^-1 (0.6752)`
? = 42.47°
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