Math 9 - Sine Law
The Law of Sine
Using Right Triangle Trigonometry, prove the Law of Sines.
Refer to Triangle ABC above. The first one is done for you.
Tick the first box.
We can use the sine function to find an expression for h in the red triangle.
Hint: Solve for Sin A.
We have,
cross multiply, we get
Now, tick the second box. Use the sine function to find the length of h in the orange triangle.
Hint: Solve for Sin C.
Eliminate h in your answers by equating them together. Divide both sides by ac What do you get?
Refer to Triangle ABC above. The third box is done for you. Tick the third box. We can use the sine function to find an expression for k in the green triangle. Hint: Solve for Sin B. We have, cross multiply, we get Now, tick the fourth box. Use the sine function to find the length of k in the blue triangle. Hint: Solve for Sin C.
Eliminate k in your answers by equating them together. Divide both sides by bc What do you get?
The proportion that you just made is what we call the Law of Sines.
Now, let us observe the law of sines in action. Tick the boxes A, B, and C to show its ratios. Observe what happens.
Adjust the values of the angles and the sides by sliding the points A, B, C. What happens to the ratio of any side length to the sine of the angle opposite that side? Is this true for all?