14.1 law of sines homework answers

The Law of Sines

The Law of Sines (or Sine Rule ) is very useful for solving triangles:

a sin A = b sin B = c sin C

It works for any triangle:

And it says that:

When we divide side a by the sine of angle A it is equal to side b divided by the sine of angle B , and also equal to side c divided by the sine of angle C

Well, let's do the calculations for a triangle I prepared earlier:

a sin A = 8 sin(62.2°) = 8 0.885... = 9.04...

b sin B = 5 sin(33.5°) = 5 0.552... = 9.06...

c sin C = 9 sin(84.3°) = 9 0.995... = 9.04...

The answers are almost the same! (They would be exactly the same if we used perfect accuracy).

So now you can see that:

Is This Magic?

Not really, look at this general triangle and imagine it is two right-angled triangles sharing the side h :

The sine of an angle is the opposite divided by the hypotenuse, so:

a sin(B) and b sin(A) both equal h , so we get:

a sin(B) = b sin(A)

Which can be rearranged to:

a sin A = b sin B

We can follow similar steps to include c/sin(C)

How Do We Use It?

Let us see an example:

Example: Calculate side "c"

Now we use our algebra skills to rearrange and solve:

Finding an Unknown Angle

In the previous example we found an unknown side ...

... but we can also use the Law of Sines to find an unknown angle .

In this case it is best to turn the fractions upside down ( sin A/a instead of a/sin A , etc):

sin A a = sin B b = sin C c

Example: Calculate angle B

Sometimes there are two answers .

There is one very tricky thing we have to look out for:

Two possible answers.

This only happens in the " Two Sides and an Angle not between " case, and even then not always, but we have to watch out for it.

Just think "could I swing that side the other way to also make a correct answer?"

Example: Calculate angle R

The first thing to notice is that this triangle has different labels: PQR instead of ABC. But that's OK. We just use P,Q and R instead of A, B and C in The Law of Sines.

But wait! There's another angle that also has a sine equal to 0.9215...

The calculator won't tell you this but sin(112.9°) is also equal to 0.9215...

So, how do we discover the value 112.9°?

Easy ... take 67.1° away from 180°, like this:

180° − 67.1° = 112.9°

So there are two possible answers for R: 67.1° and 112.9° :

Both are possible! Each one has the 39° angle, and sides of 41 and 28.

So, always check to see whether the alternative answer makes sense.

• ... sometimes it will (like above) and there are two solutions
• ... sometimes it won't (like below) and there is one solution

For example this triangle from before.

As you can see, we can try swinging the "5.5" line around, but no other solution makes sense.

So this has only one solution.

Algebra 2 Common Core

By hall, prentice, chapter 14 - trigonometric identities and equations - 14-1 trigonometric identities - practice and problem-solving exercises - page 908: 24, work step by step, update this answer.

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Law of Sines

The law of sines & law of cosines are used to find the missing sides and angles in non-right triangles., law of sines is used whenever at least one side and the angle opposite of the side both have known values., law of cosines is used for all other triangles..

Solve each triangle

\(\textbf{1)}\) show answer \(\angle c=55^{\circ} \,\,\, a\approx5.611 \,\,\, b\approx7.075\) show work \(\text{step }1: \text{find the third angle} \angle c\) \(\,\,\,\,\,\,\angle a + \angle b + \angle c = 180^{\circ}\) \(\,\,\,\,\,\,50^{\circ} + 75^{\circ} + \angle c = 180^{\circ}\) \(\,\,\,\,\,\,125^{\circ} + \angle c = 180^{\circ}\) \(\,\,\,\,\,\,\angle c = 55^{\circ}\) \(\text{step }2: \text{find the first missing side, side } b\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{c}}{c}=\frac{\sin{b}}{b}\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{55}}{6}=\frac{\sin{75}}{b}\) \(\,\,\,\,\,\,\displaystyle \frac{0.819152}{6}=\frac{0.965926}{b}\) \(\,\,\,\,\,\,0.819152 \cdot b=0.965926 \cdot 6\) \(\,\,\,\,\,\,0.819152b=5.79555\) \(\,\,\,\,\,\,b\approx7.075\) \(\text{step }3: \text{find the second missing side, side } a\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{c}}{c}=\frac{\sin{a}}{a}\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{55}}{6}=\frac{\sin{50}}{a}\) \(\,\,\,\,\,\,\displaystyle \frac{0.819152}{6}=\frac{0.766044}{a}\) \(\,\,\,\,\,\,0.819152 \cdot a=0.766044 \cdot 6\) \(\,\,\,\,\,\,a\approx5.61101\) \(\text{the answer is }\angle c=55^{\circ} \,\,\, a\approx5.611 \,\,\, b\approx7.075\), \(\textbf{2)}\) show answer \(\angle b=40^{\circ} \,\,\, b\approx6.527 \,\,\, c\approx8.794\) show work \(\text{step }1: \text{find the third angle} \angle b\) \(\,\,\,\,\,\,\angle a + \angle b + \angle c = 180^{\circ}\) \(\,\,\,\,\,\,80^{\circ} + \angle b + 60^{\circ} = 180^{\circ}\) \(\,\,\,\,\,\,\angle b + 140^{\circ} = 180^{\circ}\) \(\,\,\,\,\,\,\angle b = 40^{\circ}\) \(\text{step }2: \text{find the first missing side, side } c\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{a}}{a}=\frac{\sin{c}}{c}\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{80}}{10}=\frac{\sin{60}}{c}\) \(\,\,\,\,\,\,\displaystyle \frac{.984808}{10}=\frac{.866025}{c}\) \(\,\,\,\,\,\,.984808 \cdot c=.866025 \cdot 10\) \(\,\,\,\,\,\,c \approx 8.794\) \(\text{step }3: \text{find the second missing side, side } b\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{a}}{a}=\frac{\sin{b}}{b}\) \(\,\,\,\,\,\,\displaystyle \frac{\sin{80}}{10}=\frac{\sin{40}}{b}\) \(\,\,\,\,\,\,\displaystyle \frac{.984808}{10}=\frac{.642787609}{b}\) \(\,\,\,\,\,\,.984808b=.642787609 \cdot 10\) \(\,\,\,\,\,\,b\approx6.527\) \(\text{the answer is }\angle b=40^{\circ} \,\,\, b\approx6.527 \,\,\, c\approx8.794\), \(\textbf{3)}\) show answer, \(\textbf{4)}\) show answer, see related pages\(\), \(\bullet\text{ geometry homepage}\) \(\,\,\,\,\,\,\,\,\text{all the best topics…}\), \(\bullet\text{ triangle calculator (calculator.net)}\) \(\,\,\,\,\,\,\,\,\), \(\bullet\text{ right triangle trigonometry}\) \(\,\,\,\,\,\,\,\,\sin{(x)}=\displaystyle\frac{\text{opp}}{\text{hyp}}…\), \(\bullet\text{ angle of depression and elevation}\) \(\,\,\,\,\,\,\,\,\text{angle of depression}=\text{angle of elevation}…\), \(\bullet\text{ convert to radians and to degrees}\) \(\,\,\,\,\,\,\,\,\text{radians} \rightarrow \text{degrees}, \times \displaystyle 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Law of Cosines: The Law of Cosines is another formula that relates the sides and angles of a triangle. It is used to solve problems involving triangles when we know the lengths of all three sides of the triangle, or when we know two sides and the included angle. The Law of Cosines is often used in conjunction with the Law of Sines to solve more complex problems involving triangles. Spherical trigonometry: Spherical trigonometry is the study of triangles on the surface of a sphere. It is used in fields such as astronomy and geography to calculate distances and angles between points on a sphere. The Law of Sines is a fundamental concept in spherical trigonometry and is used to solve problems involving triangles on a sphere. Triangulation: Triangulation is a method of finding the location of an object using the principles of geometry and trigonometry. It involves measuring the angles between the object and two known points, and using the Law of Sines to calculate the distance to the object. Triangulation is used in a variety of fields, including surveying, navigation, and telecommunications. Vector calculus: Vector calculus is a branch of mathematics that deals with vector-valued functions and their derivatives. It is used to analyze and describe physical phenomena, such as fluid flow and electric and magnetic fields. The Law of Sines is used in vector calculus to solve problems involving triangles and vector fields. Differential equations: Differential equations are equations that describe how a variable changes over time. They are used to model physical systems and predict how they will behave. The Law of Sines is used in the solution of certain types of differential equations, such as those involving triangles and oscillations.

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14.1 Law of Sines Practice

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• 1. Multiple Choice Edit 5 minutes 1 pt Identify the Law of Sines Sin 2 x + Cos 2 x =1 Sin(A)/a = Sin(B)/b = Sin(C)/c Sin(A)Sin(B)Sin(C) = 1 Soh-Cah-Toa

Use the Law of Sines to solve for a

Solve the triangle Given: A=90 ° C=40 ° c=4

a=10 b=60 ° b=6

a=5 B=65.5 ° b=9

a=10 B=50 ° b=9

a=6 B=50 ° b=5

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PLEASE HELP!!! MATH 14.1 The law of sines

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➟ In a triangle, side “a” divided by the sine of angle A is equal to the side “b” divided by the sine of angle B is equal to the side “c” divided by the sine of angle C.

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