Law of Sines

To find the angles of a triangle, we use the Law of sines. So it is useful for solving triangles of any types: Obtuse, Acute or Right-angled Triangle.

  • It can be used to compute the other sides of a triangle when two angles and one side is given.

Or

  • When two sides and one non included angle is given.
    Law of Sines

    \(\frac{a}{Sin A}=\frac{B}{Sin B}=\frac{c}{Sin C}\)

Law of Sine

So, we use the Sine rule to find unknown lengths or angles of the triangle. It is also called as Sine Rule, Sine Law or Sine Formula. Check out the law of sines formula to know more about it.

Law of Sine Formula:

  • a / Sin A= B / Sin B= c / Sin C
  • a: b: c = Sin A : Sin B : Sin C
  • a/b = Sin A / Sin B
  • Similarly, b / c = Sin B/ Sin C

It denotes that if we divide side a by the Sine of A, it is equal to the division of side b by the Sine of B and also equal to the division of side c by Sine of C.

Or

The sides of a triangle are to one another in the same ratio as the sines of their opposite angles.

Here, Sin A is a number and a is the length.

Law of Sine Proof:

We need a right-angled triangle to prove the above as the trigonometric functions are mostly defined in terms of this type of triangle only.

Given: A ABC.

Construction: Draw a perpendicular, CD AB. Then CD = h is the height of the triangle. h, separates the ABC in two right-angled triangles, CDA and CDB.

To Show: a / b = Sin A / Sin B

Proof: In the  CDA,

           Sin A= h/b

           And in  CDB,

           Sin B = h/a

Therefore, Sin A / Sin B = h / a / h / b= h / b × a / h= a/b

And we proved it.

Similarly, we can prove, Sin B/ Sin C= b / c and so on for any pair of angles and their opposite sides.

Practice Example for Law of Sine Formula:

Question: Solve   PQR in which P = 63.5° and Q = 51.2°  and r = 6.3 cm.

Solution: Let’s Calculate the third angle :

  • B = 180° – 63.5°– 51.2 °= 65.3°

Now, Let’s calculate the sides:

6.3 / Sin 65.3 = p / Sin 63.5 (BC= p)

p = (6.3 × Sin 63.5) / Sin 65.3

p= 6.21 cm approx(From the table below)

Similarly, 6.3 / Sin 65.3 = q / Sin 51.2 (q= AB)

q = (6.3 × Sin51.2) / Sin 65.3

q= 5.40 cm (From the table below)

C= 65.3°

p=6.21 cm

q=5.40 cm

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Trigonometric Table

θ sin θ cos θ tan θ cot θ sec θ csc θ  
cos θ sin θ cot θ tan θ csc θ sec θ
0 1 0 …….. 1 …….. 90°
0.017 1 0.017 57.29 1 57.299 89°
0.035 0.999 0.035 28.64 1.001 28.654 88°
0.052 0.999 0.052 19.08 1.001 19.107 87°
0.07 0.998 0.07 14.3 1.002 14.336 86°
0.087 0.996 0.087 11.43 1.004 11.474 85°
0.105 0.995 0.105 9.514 1.006 9.567 84°
0.122 0.993 0.123 8.144 1.008 8.206 83°
0.139 0.99 0.141 7.115 1.01 7.185 82°
0.156 0.988 0.158 6.314 1.012 6.392 81°
  sin θ cos θ tan θ cot θ sec θ csc θ  
cos θ sin θ cot θ tan θ csc θ sec θ
10° 0.174 0.985 0.176 5.671 1.015 5.759 80°
11° 0.191 0.982 0.194 5.145 1.019 5.241 79°
12° 0.208 0.978 0.213 4.705 1.022 4.81 78°
13° 0.225 0.974 0.231 4.331 1.026 4.445 77°
14° 0.242 0.97 0.249 4.011 1.031 4.134 76°
15° 0.259 0.966 0.268 3.732 1.035 3.864 75°
16° 0.276 0.961 0.287 3.487 1.04 3.628 74°
17° 0.292 0.956 0.306 3.271 1.046 3.42 73°
18° 0.309 0.951 0.325 3.078 1.051 3.236 72°
19° 0.326 0.946 0.344 2.904 1.058 3.072 71°
  sin θ cos θ tan θ cot θ sec θ csc θ  
cos θ sin θ cot θ tan θ csc θ sec θ
20° 0.342 0.94 0.364 2.747 1.064 2.924 70°
21° 0.358 0.934 0.384 2.605 1.071 2.79 69°
22° 0.375 0.927 0.404 2.475 1.079 2.669 68°
23° 0.391 0.921 0.424 2.356 1.086 2.559 67°
24° 0.407 0.914 0.445 2.246 1.095 2.459 66°
25° 0.423 0.906 0.466 2.145 1.103 2.366 65°
26° 0.438 0.899 0.488 2.05 1.113 2.281 64°
27° 0.454 0.891 0.51 1.963 1.122 2.203 63°
28° 0.469 0.883 0.532 1.881 1.133 2.13 62°
29° 0.485 0.875 0.554 1.804 1.143 2.063 61°
  sin θ cos θ tan θ cot θ sec θ csc θ  
cos θ sin θ cot θ tan θ csc θ sec θ
30° 0.5 0.866 0.577 1.732 1.155 2 60°
31° 0.515 0.857 0.601 1.664 1.167 1.972 59°
32° 0.53 0.848 0.625 1.6 1.179 1.887 58°
33° 0.545 0.839 0.649 1.54 1.192 1.836 57°
34° 0.559 0.829 0.675 1.483 1.206 1.788 56°
35° 0.574 0.819 0.7 1.428 1.221 1.743 55°
36° 0.588 0.809 0.727 1.376 1.236 1.701 54°
37° 0.602 0.799 0.754 1.327 1.252 1.662 53°
38° 0.616 0.788 0.781 1.28 1.269 1.624 52°
39° 0.629 0.777 0.81 1.235 1.287 1.589 51°
  sin θ cos θ tan θ cot θ sec θ csc θ  
cos θ sin θ cot θ tan θ csc θ sec θ
40° 0.643 0.766 0.839 1.192 1.305 1.556 50°
41° 0.656 0.755 0.869 1.15 1.325 1.524 49°
42° 0.669 0.743 0.9 1.111 1.346 1.494 48°
43° 0.682 0.731 0.933 1.072 1.367 1.466 47°
44° 0.695 0.719 0.966 1.036 1.39 1.44 46°
45° 0.707 0.707 1 1 1.414 1.414 45

Practise This Question

Which of these puzzle pieces are symmetrical?