Efficiently Solve the Triangle Using the Law of Sines

Law of Sines	Substitute	Simplify
$\frac{ℓ}{s i n L} = \frac{d}{s i n D}$	$\frac{ℓ}{s i n 4 1 ^{\circ}} = \frac{5 3 0}{s i n 8 0 ^{\circ}}$	$ℓ \approx 353$
$\frac{e}{s i n E} = \frac{d}{s i n D}$	$\frac{e}{s i n 5 9 ^{\circ}} = \frac{5 3 0}{s i n 8 0 ^{\circ}}$	$e \approx 461$

Law of Sines

Substitute

Simplify

\frac{ℓ}{s i n L} = \frac{d}{s i n D}

\frac{ℓ}{s i n 4 1 ^{\circ}} = \frac{5 3 0}{s i n 8 0 ^{\circ}}

ℓ \approx 353

\frac{e}{s i n E} = \frac{d}{s i n D}

\frac{e}{s i n 5 9 ^{\circ}} = \frac{5 3 0}{s i n 8 0 ^{\circ}}

e \approx 461

For the triangle in the diagram, find the value of $y .$ Round the answer to one decimal place.

Triangle ABC with one angle of 32 degrees and one of 50 degrees

Notice that we have expressions for the opposite side of ∠ C and the opposite side of ∠ A. Therefore, if we can figure out m∠ A, we would be able to use the Law of Sines to write an equation that contains y.

Law of Sines |- In any triangle, the ratio between the length of a side and the sine value of the side's opposite angle is constant.

Since we know the measures of two angles, we can by the Interior Angles Theorem determine the measure of ∠ A. 50^(∘)+32^(∘)+m∠ A=180^(∘) ⇓ m∠ A=98^(∘) Let's add the measure of ∠ A to the diagram.

Now we can determine y by using the Law of Sines. 5y-20/sin98^(∘)=3y+3/sin 50^(∘) Let's solve for y.

The value of y is about 21.3.

Dominika and Tadeo have been given the following information about a triangle.

m ∠ X = 2 5^{\circ} Y Z = 2.8 inches X Z = 6.3 inches

Dominika tells Tadeo, "The triangle I drew with those measures has angle

Z

as its greatest angle."
Tadeo responds, "Well, I drew a triangle with the same measures, but angle

Y

is its greatest angle."
Who, if either, is correct?

To determine the validity of their claims, we must determine the angle measures of △ XYZ. Let's start by sketching a triangle that fits the information given in the exercise.

The question is what measure ∠ Y and ∠ Z can have. Let's start by determining m∠ Y by using the Law of Sines.

When we know m∠ Y, we can determine the last angle as well by using the Interior Angles Theorem. m∠ Z + 72^(∘)+ 25^(∘)=180^(∘) ⇓ m∠ Z=83^(∘) If we add all of the angle measures to the diagram, it would appear that Dominika is correct.

However, recall the relationship sin θ = sin(180^(∘) -θ). This means there is a second angle that gives the same sine value as 72^(∘). sin 72^(∘) = sin(180^(∘) -72^(∘)) ⇓ sin 72^(∘) = sin 108^(∘) If the circumstances are right, ∠ Y could also be 108^(∘). Since the opposite side of ∠ Y is greater than the opposite side of ∠ X, we know that there is a second triangle that fulfills the given characteristic.

We are now able to see that both Dominika and Tadeo are right. How they chose to draw their respective triangle affected their statements.

One of the grandest lighthouses in the world is the Follyhead Lighthouse in northern Scotland. In spite of its remote location it is visited by more than $100000$ people each year. Since many of these visitors arrive by cruise ship, an aerial lift nicknamed $130 - 130$ has been built from the shore to the top of the cliff. The nickname $130 - 130$ comes from the fact that the aerial lift is $130$ meters long and that it makes an angle of $13 0^{\circ}$ to the horizon. On a boat some distance from the shore Tearrik is standing at the bow and is watching the lighthouse and the cliff it stands on.

Tearrik measures the angle of elevation to the top of the lighthouse to be $2 8^{\circ}$ and the angle of elevation to the base of the lighthouse to $1 9^{\circ} .$

How far away from the shore is Tearrik? Round to the nearest meter.

What is the height of the lighthouse? Round to the nearest meter.

Let's first draw a diagram that represents the situation. Remember, an angle of elevation is the angle between the horizon and the line of sight to an object. Notice that the angle between the two lines of sight is the difference of the given angles of elevation. 28^(∘)-19^(∘)=9^(∘) We will label the distance between Tearrik and the shore as d.

To find the value of d, we will use the Law of Sines. To do so, first we need to know the angle that is opposite of d. Since we have the measures of two angles in the triangle, we can use the Interior Angles Theorem to find the measure of the angle opposite d. m∠ θ +130^(∘) +19^(∘) = 180^(∘) ⇓ m∠ θ = 31^(∘) Let's add this to the diagram.

Now we have enough information to apply the Law of Sines to our triangle. d/sin 31^(∘)=130/sin 19^(∘) Let's solve for d.

The distance between Tearrik and the shore is about 206 meters.

Now we want to find the height of the lighthouse. We will label this h in our diagram. We will also label the line of sight to the base of the lighthouse x.

We can find the distance x by using the Law of Sines. x/sin130^(∘)=130/sin 19^(∘) Let's solve this equation.

Let's add this value to the diagram. We will also draw a line from the top of the lighthouse that is also parallel to the horizon. Tearrik's line of sight to the top of the lighthouse now creates a pair of alternate interior angles. Since the line we drew is parallel to horizon by the Alternate Interior Angles Theorem the alternate interior angles are congruent.

We can now use that we have two complementary angles at the top of the lighthouse to find the measure of the angle between the lighthouse and the line of sight. α + 28^(∘) =90^(∘) ⇔ α= 62 ^(∘) With this we know two angle measures and one side for the triangle created by the two lines of sight.

To find the height of the lighthouse we can use the Law of Sines. h/sin 9^(∘)=305.88304.../sin 62^(∘) Let's solve for h.

The height of the lighthouse is about 54 meters.

Is there a triangle

A B C

with the following characteristics?

m ∠ A = 3 9^{\circ} A C = 4 cm B C = 2.5 cm

Please explain your reasoning.

Let's start by drawing a triangle.

Let's in this drawing write the angle measure and the side lengths we were given.

But is this even a possible triangle? We can figure this out by using the Law of Sines. If the triangle is possible, we should be able to solve for m∠ B.

If we attempt to calculate this on our graphing calculator, we will get an error.

But why do we get this error? Well, it is because the quotient of 4 sin 39^(∘) 2.5 is greater than 1. 4 sin 39^(∘)/2.5=1.00691... The sine ratio has a maximum value of 1. We can better understand why this is by drawing y=1.00691 and the unit circle in a coordinate plane. Remember, the sine ratio is measured on the vertical axis in the unit circle.

If we were to try to draw a triangle with the given dimensions, the diagram represents what it would look like.

As we can see, this is not a complete triangle. Therefore, there is not a triangle with the given characteristics.

Solving Triangles Using the Law of Sines

Catch-Up and Review

Investigate Triangles With Two Sides and an Angle Known

Relationship Between a Triangle's Angles and Their Opposite Sides

Law of Sines

Proof

Acute Triangles

Obtuse Triangles

Using the Law of Sines To Find the Side of a Triangle

Hint

Solution

Practice Finding Sides Using the Law of Sines

Using the Law of Sines To Find Angles of a Triangle

Hint

Solution

Practice Finding Angles Using the Law of Sines

Using the Law of Sines To Find Measurements of a Triangle

Hint

Solution

Perimeter

Area

The Ambiguous Case of the Law of Sines

Calculating an Angle of a Triangle Using the Law of Sines

Hint

Solution

Solving Triangles Using the Law of Sines

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