Understanding When to Use the Cosine Law
- The sine law requires at least one known pair of opposite angles and sides.
- If such pairs are not available, particularly when you have two sides and the included angle, sine law cannot be applied.
- In these cases, the cosine law becomes the appropriate tool. For a thorough foundation, consider reviewing Solving Varying Angle Problems Using Sine and Cosine Laws.
The Cosine Law Formula
- The law relates the lengths of sides of a triangle to the cosine of one of its angles.
- Three equivalent forms exist, each solving for a different side: [ a^2 = b^2 + c^2 - 2bc \cdot \cos A ] [ b^2 = a^2 + c^2 - 2ac \cdot \cos B ] [ c^2 = a^2 + b^2 - 2ab \cdot \cos C ]
Example 1: Finding a Side Length Opposite a Known Angle
- Given angle A = 67°, side b = 12, side c = 10
- Use the formula: [ a^2 = b^2 + c^2 - 2bc \cos A ]
- Calculation steps:
- [ a^2 = 144 + 100 - 240 \times \cos 67° ]
- [ a^2 ≈ 244 - 240 \times 0.3907 = 244 - 93.78 = 150.22 ]
- [ a ≈ \sqrt{150.22} ≈ 12.26 ]
- This example aligns with methods shown in How to Find Triangle Side Lengths Using Trigonometry.
Example 2: Finding an Angle Given All Three Sides
- Given sides a = 7, b = 5, c = 8
- To find angle B opposite side b, use: [ \cos B = \frac{a^2 + c^2 - b^2}{2ac} ]
- Calculation steps:
- [ \cos B = \frac{49 + 64 - 25}{2 \times 7 \times 8} = \frac{88}{112} ≈ 0.7857 ]
- [ B = \cos^{-1}(0.7857) ≈ 38.21° ]
Summary
- Use the cosine law when you have:
- Two sides and the included angle (to find the opposite side).
- All three sides (to find an unknown angle).
- The cosine law complements the sine law and covers cases where the sine law is not applicable.
Learning to apply the cosine law effectively expands your toolkit for solving various triangle-related problems in trigonometry. For a comprehensive overview on related techniques, you might also explore How to Solve Right Triangles Using Pythagorean Theorem and Trigonometry and How to Use SOHCAHTOA to Find Missing Angles in Triangles.
in this video let's learn about the klaw and get some practice with how to use [Music]
them so before we start this lesson I'd like to encourage you to watch the sign law video first if you haven't already
done so as it is usually taught before the Coan law otherwise let's Dive Right into our first example so in a situation
like this we cannot use the sign law in order to further solve the dimensions of this triangle and why is that because
the sign law is used when we have the information for at least one pair of opposite Dimensions like so along with
either another angle or side length in this case we have the information for one angle and two side lengths
but none of the provided values are opposites from each other meaning that we are unable to use the sign law to
find the other values here well this is not a problem since the cosine law is what can be used in this exact scenario
when we're provided with two side lengths and one angle in between them we can use the cosine law to find the side
length opposite to that angle so let's take take a look at what the cosine law actually looks
like so here is the cosine law and as you can see there are three different versions of the formula however they are
in essence all the same since two of the forms are just rearrangements of the cosine law to show the formula in terms
of each side so let's go ahead and choose one of them and try solving the example that we saw earlier
all right so since our objective here is to find the value of a we need to start by plugging in the angle opposite to a
which is 67° into the cosine of a and we'd get cosine of 67° here now we still have B
and C to fill in and since we have the information to the two sides besides this angle we can plug 12 and 10 into
the two B's and C's here and we'd get the following simplifying this would give us
a^2 = 144 + 100us 240 * cosine of 67° here Computing the right side gives us roughly 1
15.22 and all we need to do now is square root both sides to get a final answer of a equals roughly
12.26 great so we just said that when we have two sides and an angle in between we
should use the cosine law to find the side length that is opposite to the known angle however there is another
interesting situation where we can use the cosine law that we should probably go over before we conclude this
video let's say we were given this following triangle what if the question asks us us to find the value of angle B
how would we do this again this is definitely not the time to use the sign law since we don't have any angles to
use at all but since we're provided with the values of three sides to a triangle we can in fact use the cosine law here
as well so let's try this example to explore this idea further so since we want to find the value of angle B let's
use this version of the cosine law so we plug in five for B since it's the value opposite to the angle that
we're looking for and plug the two other side values into their respective variables and we're left with cosine of
b as our unknown so our objective is to isolate it so let's simplify this to get the following and add 49 and and 64 to
get 1133 add both sides by - 113 and divide both sides by
-112 so this gives us cine of b equal to 88 over 112 all we need to do now to get the
value of angle B is to take the inverse of cosine 88 over 112 and we get the angle B equals to
roughly 38.2 1° so there's our answer awesome so to sum it all up we can use the cosine law whenever we're
provided with either two side values with an angle in between to find the value of the opposite side to the angle
like so or when we have the values of all three sides to a triangle to find any one angle in this triangle so now
you know how to use the cosign law to help you solve some questions hope you guys enjoyed this video and we'll see
you in the next one
You should use the cosine law when you don't have a known pair of opposite angles and sides, specifically when you have two sides and the included angle, which makes the sine law inapplicable. The cosine law is ideal for calculating a side opposite a known angle or an angle opposite a known side in such cases.
The cosine law formula relates the lengths of sides of a triangle with the cosine of one of its angles. For example, to find side a opposite angle A, use: a² = b² + c² - 2bc × cos A. Plug in the known side lengths b and c, and angle A, then solve for a by calculating the square root of the result.
If sides a, b, and c are known, to find angle B opposite side b, apply the formula: cos B = (a² + c² - b²) / (2ac). Calculate the value on the right, then determine angle B by finding the inverse cosine (cos⁻¹) of that value, which gives the angle in degrees.
Yes. For instance, if angle A = 67°, side b = 12, and side c = 10, use the formula a² = b² + c² - 2bc × cos A. Calculate: a² = 144 + 100 - 240 × cos 67° ≈ 244 - 93.78 = 150.22. Then, a ≈ √150.22 ≈ 12.26, which gives the length of side a.
The cosine law is most suitable for solving triangles when you have either two sides and the included angle to find the third side or all three sides to find an unknown angle. It is especially useful when the sine law cannot be applied due to lacking an opposite angle-side pair.
The cosine law expands your problem-solving toolkit by handling scenarios where the sine law doesn't apply, such as when two sides and their included angle are known or when all three sides are known but you need an angle. Together, sine and cosine laws cover a wider range of triangle problems effectively.
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