Assume αis opposite side a, βis opposite side b, and γis opposite side c. Solve each triangle fo

step 1 This question is about the law of signs. It asks us to solve this triangle. Now, to solve right

Assume αis opposite side a, βis opposite side b, and γis...

Assume $\alpha$ is opposite side $a, \beta$ is opposite side $b,$ and $\gamma$ is opposite side $c$. Solve each triangle for the unknown sides and angles if possible. If there is more than one possible solution, give both.
$$
\alpha=119^{\circ}, a=26, b=14
$$

Key Concepts

Law of Sines

The Law of Sines provides a relationship between the sides of a triangle and the sines of their opposite angles. It states that for any triangle, the ratio of a side's length to the sine of its corresponding angle is constant, enabling the determination of unknown sides or angles when sufficient information (such as ASA, AAS, or SSA configurations) is available.

Ambiguous Case (SSA)

The ambiguous case occurs in scenarios where two sides and a non-included angle (SSA) are known. In such cases, the given information may correspond to two distinct triangles, one triangle, or no triangle at all. Resolving this ambiguity involves careful application of the Law of Sines and considering the range of values possible for the sine function.

Triangle Angle Sum

The triangle angle sum theorem asserts that the interior angles of any triangle always add up to 180 degrees. This fundamental property is crucial for solving triangles, as determining two angles allows the calculation of the third angle by subtracting the sum of the known angles from 180 degrees.

Solving Triangles

Solving triangles involves the process of finding all unknown sides and angles based on a set of given measurements. Depending on the type of provided information—such as side-side-angle (SSA), angle-side-angle (ASA), or side-side-side (SSS)—different methods like the Law of Sines, the Law of Cosines, and trigonometric identities are employed to arrive at a complete solution.

Transcript

00:02 This question is about the law of signs.

00:05 It asks us to solve this triangle.

00:10 Now, to solve right angle to triangles, of course, we can't use pythagorean theorem or trig ratios directly, because those only work for right -angled triangles.

00:28 For non -right -angled triangles, we have to either use the law of signs or the law of cosines.

00:34 Now, the law of signs is simpler, so you should try to use that whenever you can.

00:38 And i can tell you when that is.

00:42 The law of cosines should only be used when you are given just the three side lengths or just two side lengths and the angle in between them.

00:54 Only in those cases do you need to use the cosine law or the law of cosines.

00:59 In all other cases, as long as you're given enough information, you can solve using the law of signs.

01:06 Okay, now we get on with a question.

01:09 We want to find the values of c, the angle, beta, and gamma, where beta and gamma are angles, and c is a side length.

01:22 And the law of science, recall, looks like this.

01:44 Now, how do we use that to find one of these three quantities? well, first of all, we need to know, well, to use one equation, we need to know three of the four quantities so that we can find the fourth.

02:01 And so we need to know one pair and an additional piece of information.

02:06 So aside an angle pair, such as a and alpha, and either one other side length or one other angle.

02:17 Now, if we take a look, we don't have the beta b pair because beta is unknown.

02:23 We don't know the gamma -c pair because neither of them are known.

02:27 We only have, the only known pair is the a alpha pair...

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