Grade 10

Advanced

10 min

Lesson 9 of 12

Law of Sines and Law of Cosines

Not Started

Extends trigonometric principles to non-right (oblique) triangles.

If you were a search-and-rescue pilot trying to locate a signal between two radio towers, how would you calculate the exact distance to the survivor without a single right angle in sight?

Learning Objectives

1.Apply the Law of Sines to solve AAS and ASA triangle cases
2.Utilize the Law of Cosines for SAS and SSS scenarios
3.Identify and solve the 'Ambiguous Case' (SSA) where multiple triangles may exist

The Law of Sines: The Power of Proportions

When we move beyond right-angled triangles, we need new tools. The Law of Sines establishes a beautiful relationship between the sides of a triangle and the sines of their opposite angles. It states that the ratio of each side length to the sine of its opposite angle is constant for all three sides:

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

This law is your primary tool when you know an Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) configuration. Essentially, if you have at least one 'complete pair' (a side and its opposite angle), you can find any other missing piece of the triangle using simple cross-multiplication.

Given a triangle where $\angle A = 40^\circ$ , $\angle B = 60^\circ$ , and side $a = 10$ , find side $b$ . 1. Identify the knowns: $A = 40^\circ, a = 10, B = 60^\circ$ . 2. Set up the Law of Sines: $\frac{10}{\sin 40^\circ} = \frac{b}{\sin 60^\circ}$ . 3. Isolate $b$ : $b = \frac{10 \cdot \sin 60^\circ}{\sin 40^\circ}$ . 4. Calculate: $b \approx \frac{10 \cdot 0.866}{0.643} \approx 13.47$ .

Quick Check

To use the Law of Sines, what is the minimum 'pair' of information you must have?

Look at the structure of the ratio

\frac{a}{\sin A}

Answer

You must have at least one side and its corresponding opposite angle.

The Law of Cosines: The Generalized Pythagoras

What happens if you don't have an opposite pair? If you only know all three sides (SSS) or two sides and the included angle (SAS), the Law of Sines fails. Enter the Law of Cosines. It is essentially the Pythagorean Theorem adjusted for non-right angles:

a^2 = b^2 + c^2 - 2bc \cos A

Notice that if

A = 90^\circ

, then

\cos 90^\circ = 0

, and the formula collapses back into

a^2 = b^2 + c^2

. This law allows us to 'bridge the gap' between two sides to find the third, or to find an angle when only side lengths are known.

A hiker walks 5 km East, then turns $60^\circ$ North of East and walks 8 km. How far is she from her starting point? 1. This is a SAS case with $b=5, c=8$ , and included angle $A=120^\circ$ (since $180-60=120$ for the interior angle). 2. Apply formula: $a^2 = 5^2 + 8^2 - 2(5)(8) \cos 120^\circ$ . 3. $a^2 = 25 + 64 - 80(-0.5)$ . 4. $a^2 = 89 + 40 = 129$ . 5. $a = \sqrt{129} \approx 11.36$ km.

Quick Check

Which law would you use first if you were given the lengths of all three sides (SSS)?

Can you form a ratio with an angle if no angles are given?

Answer

The Law of Cosines.

The Ambiguous Case: The SSA Trap

The Side-Side-Angle (SSA) case is known as the 'ambiguous case' because knowing two sides and a non-included angle doesn't always define a unique triangle. Depending on the length of the side opposite the given angle ( $a$ ) relative to the height of the potential triangle ( $h = b \sin A$ ), there could be zero, one, or two possible triangles. If $a < h$ , no triangle exists. If $a > b$ , only one triangle exists. However, if $h < a < b$ , two different triangles can be formed—one acute and one obtuse. Always check the sine value: if $\sin B > 1$ , the triangle is impossible!

In $\triangle ABC$ , $A=30^\circ, a=6, b=10$ . Find $\angle B$ . 1. Use Law of Sines: $\frac{6}{\sin 30^\circ} = \frac{10}{\sin B}$ . 2. $\sin B = \frac{10 \cdot 0.5}{6} = \frac{5}{6} \approx 0.833$ . 3. $B_1 = \arcsin(0.833) \approx 56.4^\circ$ . 4. Check for second triangle: $B_2 = 180^\circ - 56.4^\circ = 123.6^\circ$ . 5. Verify $B_2$ : $123.6^\circ + 30^\circ < 180^\circ$ . Since it fits, both $56.4^\circ$ and $123.6^\circ$ are valid solutions.

Key Takeaways

1Use the Law of Sines ( $\frac{a}{\sin A} = \frac{b}{\sin B}$ ) for AAS or ASA cases where an opposite pair is known.
2Use the Law of Cosines ( $a^2 = b^2 + c^2 - 2bc \cos A$ ) for SAS or SSS cases.
3The SSA case is 'ambiguous' and requires checking if zero, one, or two triangles are possible.
4When solving for an angle using Law of Sines, always check if the supplementary angle ( $180^\circ - \theta$ ) could also form a valid triangle.

Quick Check

Multiple Choice

In a triangle where you know all three side lengths, which law is most efficient to find the first angle?

Multiple Choice

If $\sin B = 1.2$ after performing Law of Sines calculations, what does this indicate?

True/False

The Law of Cosines can be used to find a missing side if you know two sides and the angle between them (SAS).

What's Next?

Review Tomorrow

Tomorrow morning, try to write down the Law of Cosines from memory and explain why it looks like the Pythagorean Theorem.

Practice Activity

Find a map and pick three cities. Use their coordinates to estimate the distances (sides) and use the Law of Cosines to find the angles of the triangle they form.

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Law of Sines and Law of Cosines

Learning Objectives

The Law of Sines: The Power of Proportions

The Law of Cosines: The Generalized Pythagoras

The Ambiguous Case: The SSA Trap

Key Takeaways

Quick Check

What's Next?

Law of Sines and Law of Cosines

Learning Objectives

The Law of Sines: The Power of Proportions

The Law of Cosines: The Generalized Pythagoras

The Ambiguous Case: The SSA Trap

Key Takeaways

Quick Check

What's Next?