Proving A Sin A = B Sin B = C Sin C

by Alex Braham 36 views

Hey everyone! Today, we're diving deep into a super important concept in trigonometry: the Sine Rule. You might have seen it as a/sin A = b/sin B = c/sin C, or maybe written as a sin B = b sin A, and you're probably wondering, "Okay, but why is this true? How do we actually prove it?" Well, you've come to the right place, guys! We're going to break down the proof step-by-step, making it as clear as possible. Understanding this proof is key to unlocking a whole bunch of triangle problems and feeling confident when you see it pop up in your textbooks or exams. So, grab your favorite beverage, get comfy, and let's get this done!

Understanding the Sine Rule

Before we jump into the proof, let's make sure we're all on the same page about what the Sine Rule actually is and what it applies to. The Sine Rule is a fundamental relationship that connects the lengths of the sides of any triangle to the sines of its opposite angles. That's right, any triangle – whether it's acute, obtuse, or even right-angled! This makes it incredibly versatile. In a triangle ABC, we conventionally label the side opposite angle A as 'a', the side opposite angle B as 'b', and the side opposite angle C as 'c'. The Sine Rule states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles. Mathematically, this is expressed as:

a / sin A = b / sin B = c / sin C

This also means that a / sin A = b / sin B, b / sin B = c / sin C, and a / sin A = c / sin C. Another common way you'll see it written, which is derived directly from the above, is a / b = sin A / sin B. This form is super handy when you're trying to find a missing side or angle when you have a ratio of sides and angles. The beauty of the Sine Rule is that it works for all triangles, not just right-angled ones. This is a huge advantage over basic trigonometric ratios (SOH CAH TOA) which are limited to right triangles. So, when you're faced with a triangle where you know two angles and a side, or two sides and an angle opposite one of them, the Sine Rule is likely your best friend. We'll be using this fundamental relationship to build our proof, so keep it firmly in mind!

The Proof: Constructing an Altitude

To prove the Sine Rule, the most common and intuitive method involves constructing an altitude within the triangle. Let's consider a general triangle ABC, with sides a, b, and c opposite angles A, B, and C, respectively. We want to show that a / sin A = b / sin B = c / sin C. To do this, we'll focus on proving that a / sin A = b / sin B first, and then we'll extend it to include c / sin C. This is a classic approach in geometry and trigonometry proofs, and it really helps to visualize the relationships involved. So, let's draw our triangle ABC.

Now, from vertex C, let's drop an altitude (a perpendicular line) to the opposite side AB. Let's call the point where this altitude meets AB as D. Let the length of this altitude, CD, be 'h'. This construction divides our original triangle ABC into two smaller right-angled triangles: triangle ADC and triangle BDC. This is a crucial step because, as we know, basic trigonometric ratios are defined for right-angled triangles. So, by creating these two right-angled triangles, we can now apply SOH CAH TOA to them. This is where the magic starts to happen, and we'll see how the Sine Rule emerges from these basic principles. Remember, the goal is to relate the sides (a, b, c) to the sines of the angles (A, B, C), and this altitude is our key to doing just that. We are essentially breaking down a general triangle problem into simpler, solvable right-angled triangle problems. This is a very common strategy in mathematics – simplify the complex by breaking it down into manageable parts. So, keep your eyes peeled as we work through these smaller triangles and connect them back to our original triangle ABC and the Sine Rule we aim to prove.

Case 1: Acute Triangle

Let's start with an acute triangle, where all angles are less than 90 degrees. In our triangle ABC, we've dropped the altitude CD = h from C to AB. Now, let's look at the two right-angled triangles formed: triangle ADC and triangle BDC.

In triangle ADC (which is right-angled at D): We can use the sine function. Remember SOH CAH TOA? Sine is Opposite over Hypotenuse. In triangle ADC, the side opposite angle A is CD (which is our altitude 'h'), and the hypotenuse is AC (which is side 'b'). So, we have:

sin A = h / b

Rearranging this to find 'h', we get:

h = b sin A (Equation 1)

Now, let's look at triangle BDC (which is also right-angled at D): Here, the side opposite angle B is CD ('h'), and the hypotenuse is BC (which is side 'a'). So, we have:

sin B = h / a

Rearranging this to find 'h', we get:

h = a sin B (Equation 2)

Now, here's the really cool part! Since both Equation 1 and Equation 2 are equal to 'h', we can set them equal to each other:

b sin A = a sin B

If we divide both sides by sin A and sin B (assuming neither angle is 0 or 180 degrees, which is true for any triangle), we get:

b / sin B = a / sin A

Voila! We've just proven one part of the Sine Rule for an acute triangle! It shows that the ratio of a side to the sine of its opposite angle is the same for sides 'a' and 'b'. We can follow the exact same procedure by dropping an altitude from vertex A to side BC, or from vertex B to side AC, to show that b / sin B = c / sin C and a / sin A = c / sin C. For example, dropping an altitude from A to BC would create right triangles where we could relate side 'a' to 'sin A' and sides 'b' and 'c' to their respective angles, ultimately linking all three ratios. So, for acute triangles, the Sine Rule holds true!

Case 2: Obtuse Triangle

What about obtuse triangles, where one angle is greater than 90 degrees? Does the Sine Rule still work? Absolutely! Let's see how the proof adapts. Consider a triangle ABC where angle A is obtuse. Again, we drop an altitude CD = h from vertex C to the line containing side AB. Note that since angle A is obtuse, the point D will lie outside the triangle, on the extension of side AB.

This construction still gives us right-angled triangles, but we need to be careful with our angles. We have right-angled triangle ADC and right-angled triangle BDC.

In right-angled triangle ADC: Angle CAD is the exterior angle to angle A of triangle ABC. The interior angle we're interested in is angle CAB, which is A. However, in triangle ADC, the angle at A is actually 180° - A (since A is obtuse, 180° - A will be acute). The sine of an angle and the sine of its supplement are equal (i.e., sin(θ) = sin(180° - θ)). Therefore, sin(A) = sin(180° - A).

In triangle ADC, the side opposite angle (180° - A) is h, and the hypotenuse is b. So, sin(180° - A) = h / b. Since sin(180° - A) = sin A, we have:

sin A = h / b

Rearranging for h:

h = b sin A (Equation 3)

Now, let's look at right-angled triangle BDC: The angle at B is simply angle B of triangle ABC. The side opposite angle B is h, and the hypotenuse is a. So, sin B = h / a.

Rearranging for h:

h = a sin B (Equation 4)

Just like in the acute case, we set Equation 3 and Equation 4 equal to each other because they both equal 'h':

b sin A = a sin B

And again, dividing by sin A and sin B gives us:

b / sin B = a / sin A

So, you see? The Sine Rule holds true even for obtuse triangles! The key was understanding that sin A = sin(180° - A), which allows us to use the standard sine definition in the right-angled triangle formed by the altitude. The geometry shifts slightly with the altitude falling outside the triangle, but the trigonometric relationships remain consistent. This demonstrates the robustness of the Sine Rule and its applicability across different triangle types. We can again extend this to include 'c' by dropping other altitudes. The principle remains the same: use the altitude to create right-angled triangles and apply basic sine definitions, adjusting for obtuse angles when necessary.

Case 3: Right-Angled Triangle

Finally, let's consider the case where the triangle is right-angled. Let's say angle C is the right angle (90°). In this scenario, the Sine Rule simplifies beautifully, and we can see it holds directly from the definitions of sine in a right triangle.

In a right-angled triangle ABC, with angle C = 90°:

  • Side 'a' is opposite angle A.
  • Side 'b' is opposite angle B.
  • Side 'c' is the hypotenuse (opposite the right angle C).

Let's look at the ratios:

  • For a / sin A: In a right triangle, sin A = Opposite / Hypotenuse = a / c. So, a / sin A = a / (a / c) = a * (c / a) = c.

  • For b / sin B: In a right triangle, sin B = Opposite / Hypotenuse = b / c. So, b / sin B = b / (b / c) = b * (c / b) = c.

  • For c / sin C: Since angle C = 90°, sin C = sin 90° = 1. So, c / sin C = c / 1 = c.

As you can see, in a right-angled triangle, all three ratios are equal to the length of the hypotenuse 'c'! Therefore:

a / sin A = b / sin B = c / sin C = c

This confirms that the Sine Rule is indeed true for right-angled triangles as a special case. It's always satisfying when a general rule simplifies nicely for specific cases, confirming its validity. This also provides a quick sanity check for the Sine Rule: if you ever have a right triangle problem, you can expect the ratio to be equal to the hypotenuse if one of the angles is 90 degrees!

Conclusion: The Power of the Sine Rule

So there you have it, guys! We've successfully proven the Sine Rule, a / sin A = b / sin B = c / sin C, for acute, obtuse, and right-angled triangles using the fundamental concept of altitudes and basic trigonometric definitions. This rule is an absolute game-changer when you're working with triangles that aren't right-angled, or when you don't have enough information to use Pythagoras' theorem or SOH CAH TOA directly.

Key Takeaways:

  • The Sine Rule relates the sides of any triangle to the sines of their opposite angles.
  • The proof relies on constructing an altitude to create right-angled triangles.
  • It works for acute, obtuse, and right-angled triangles.
  • The common form is a / sin A = b / sin B = c / sin C.

Understanding this proof not only solidifies your knowledge of trigonometry but also equips you with a powerful tool for solving a vast array of geometry problems. Whether you're calculating missing sides, finding unknown angles, or proving other geometric theorems, the Sine Rule will be your trusty sidekick. Keep practicing, and you'll be using it like a pro in no time! Happy problem-solving!