Understanding Law of Cosine: Solving Triangles with Sines and Cosines

$m ∠ C$	$c^{2} = a^{2} + b^{2} - 2 a b cos C$	Relationship between $a^{2} + b^{2}$ and $c^{2}$	Conclusion
$m ∠ C < 9 0^{\circ}$	$c^{2} = a^{2} + b^{2} - 2 a b > 0 cos C$	$c^{2} < a^{2} + b^{2}$	If $∠ C$ is acute, not too many conclusions can be made. The opposite side to $∠ C$ can be the largest side, the shortest side, or none.
$m ∠ C = 9 0^{\circ}$	$c^{2} = a^{2} + b^{2} - 2 a b = 0 cos 9 0^{\circ}$	$c^{2} = a^{2} + b^{2}$	The cosine of $9 0^{\circ}$ is $0 .$ In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to $∠ C$ is the largest side of the triangle.
$9 0^{\circ} < m ∠ C < 18 0^{\circ}$	$c^{2} = a^{2} + b^{2} - 2 a b < 0 cos C$	$c^{2} > a^{2} + b^{2}$	If $∠ C$ is obtuse, its measure is greater than the measures of $∠ A$ and $∠ C .$ Therefore, its opposite side is the largest side of the triangle.

m ∠ C

c^{2} = a^{2} + b^{2} - 2 a b cos C

Relationship between

a^{2} + b^{2}

and

c^{2}

Conclusion

m ∠ C < 9 0^{\circ}

c^{2} = a^{2} + b^{2} - 2 a b > 0 cos C

c^{2} < a^{2} + b^{2}

∠ C

is acute, not too many conclusions can be made. The opposite side to

∠ C

can be the largest side, the shortest side, or none.

m ∠ C = 9 0^{\circ}

c^{2} = a^{2} + b^{2} - 2 a b = 0 cos 9 0^{\circ}

c^{2} = a^{2} + b^{2}

The cosine of

9 0^{\circ}

0 .

In this case, the Law of Cosines becomes the Pythagorean Theorem. This means that the opposite side to

∠ C

is the largest side of the triangle.

9 0^{\circ} < m ∠ C < 18 0^{\circ}

c^{2} = a^{2} + b^{2} - 2 a b < 0 cos C

c^{2} > a^{2} + b^{2}

∠ C

is obtuse, its measure is greater than the measures of

∠ A

and

∠ C .

Therefore, its opposite side is the largest side of the triangle.

	$Speed = \frac{Distance}{Time}$
Length	$S D$	$D N$
Substitution	$330 = \frac{S D}{3}$	$330 = \frac{D N}{2}$
Calculation	$S D = 990 mi$	$D N = 660 mi$

Speed = \frac{Distance}{Time}

Length

S D

D N

Substitution

330 = \frac{S D}{3}

330 = \frac{D N}{2}

Calculation

S D = 990 mi

D N = 660 mi

LaShay and Jordan want to know the distance between two locations, $A$ and $B,$ that are on opposite ends of a forest. However, they find it difficult to measure this distance because there is not a walking path through the forest. Brilliantly, they come up with the idea to make the following measurements.

Use the given information to determine the length of

\overline{A B} .

Round the final distance to the nearest meter.

The sketch can be viewed as two triangles, △ ADC and △ ABC, where we know all three sides in △ ADC and two sides in △ ABC. Let's visualize this.

Since we know all sides of △ ADC, we can use the Law of Cosine to find the measure of ∠ C. Notice that this angle is shared by both triangles.

Let's add this to the diagram and turn our attention to △ ABC.

We can use the Law of Cosine once more to find the length of AB.

The length of AB is about 180 meters.

Determine the unknown side of the triangle in inches. Round the answer to one decimal place.

a triangle ABC with sides of 3, 4, and a inches

a triangle ABC with sides of 4.1 cm, 1.8 cm, and c inches

Since we know two sides in the triangle and their included angle, we can use the Law of Cosines to solve for the unknown side.

The unknown side is about 1.6 inches.

Again, we know two sides and the included angle which means we have enough information to find the opposite side of the angle by using the Law of Cosines.

The side labeled c is about 2.8 inches.

Including everything on the yard, what is the total area of the yard?

Round the answer to the nearest square meter.

To determine the area of the yard using trigonometry, we need to know the measures of at least two sides and their included angle. From the diagram, we know the length of each of the triangle's sides. We can then use the Law of Cosines to determine the measure of any of its angles.

Now we can use the formula for calculating area of a triangle using sine.

The area is about 2927 square meters.

Determine the measure of $∠ C .$ Round the answer nearest whole number of degrees.

a triangle with sides of 3.1 cm, 5 cm, and 3 cm

Determine the measure of $∠ B$ . Round the measure to the nearest whole number of degrees.

a triangle with sides of 2.8 cm, 4 cm, and 2.2 cm

We see from the triangle that all three sides are known. Those lengths can be used to determine m∠ C using the Law of Cosines.

As in Part A, we can use the Law of Cosine to find the measure of ∠ B.

While waiting for the bus, Tiffaniqua is trying to find the side labeled $b$ in the triangle $A B C .$

triangle ABC with sides of x inches, 3 inches, and 2 inches

She attempts to solve for $b$ by using the Law of Cosine.

However, something is wrong in her calculations. Read her notes and decide which step Tiffaniqua made a mistake. Explain your reasoning.

Let's use matching colors to identify corresponding sides and vertices.

In the Law of Cosine, the side length we substitute into the left-hand side must be opposite the angle substituted into the right-hand side. Examples [-0.8em] a^2=b^2+c^2-2bccos A [1em] b^2=a^2+c^2-2accos B [1em] c^2=a^2+b^2-2abcos C However, Tiffaniqua has substituted b as if it is opposite the 36^(∘) angle when it actually is opposite of the side labeled 2. b^2= 2^2+ 3^2-2( 2)( 3)cos 36^(∘) * However, the correct method would be to use the opposite side c to the angle C as the value on the left-hand side of the equation. 2^2= b^2+ 3^2-2( b)( 3)cos 36^(∘) ✓ This implies that Step 1 was the mistake.

Solving Triangles Using the Law of Cosines

Catch-Up and Review

Investigate Distances in Space Using Trigonometry

Limitations of the Law of Sines

Law of Cosines

Proof

Proof

Knowing Two Sides and the Included Angle of a Triangle

Hint

Solution

Knowing Three Sides of a Triangle

Hint

Solution

Finding $m ∠ L$

Finding $m ∠ M$

Finding $m ∠ K$

Investigating the Cosine Ratio of Obtuse Angles

Relationship Between the Angles of a Triangle

Solving Problems Using the Law of Cosines

Hint

Solution

Finding $S D$ and $D N$

Finding $S D$

Calculating Distances in Space Using Trigonometry

Hint

Answer

Solving Triangles Using the Law of Cosines

Recommended exercises

	10 Theory slides
	10 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Solving Triangles Using the Law of Cosines

Catch-Up and Review

Proof

Proof

Hint

Solution

Hint

Solution

Finding m∠L

Finding m∠M

Finding m∠K

Hint

Solution

Finding SD and DN

Finding SD

Hint

Answer

Solving Triangles Using the Law of Cosines

Recommended exercises

Finding $m ∠ L$

Finding $m ∠ M$

Finding $m ∠ K$

Finding $S D$ and $D N$

Finding $S D$