Use the figure and the given information to solve each triangle. (figure cannot copy) a=6.00, b=

Use the figure and the given information to solve each...

Key Concepts

Triangle Angle Sum Theorem

The Triangle Angle Sum Theorem states that the sum of the interior angles of any triangle is always 180 degrees. This fundamental property allows for the determination of an unknown angle when the other two angles are known, and it serves as a cross-check for computational accuracy when solving a triangle.

Law of Cosines

The Law of Cosines provides a relationship among the three sides of a triangle and one of its angles. It is particularly useful when dealing with triangles where two sides and the included angle are known, or when all three sides are known and an angle is sought. This law generalizes the Pythagorean theorem to all types of triangles and helps calculate an unknown side or angle accurately.

Ambiguous Case (SSA)

The ambiguous case arises in triangle solving when two sides and an angle that is not included between them are known (SSA condition). In such scenarios, the provided information can lead to one, two, or no possible solutions for the triangle, making it necessary to consider multiple outcomes and apply logical constraints to determine the correct configuration.

Triangle Solving Techniques

This concept involves determining all unknown sides and angles of a triangle when given partial information. Whether through the use of trigonometric ratios in right triangles or generalized relations like the Law of Sines and Law of Cosines in oblique triangles, solving a triangle means reconstructing the complete geometric description based on provided data.

Law of Sines

The Law of Sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This relation is extremely useful for solving triangles when given enough information (such as two angles and one side or two sides and a non-included angle), and it enables one to find unknown angles or sides by creating proportions.

Transcript

00:01 When solving a right triangle, we are expected to find the three unknown values.

00:07 And keep in mind, we will give our final answers for sides to two decimal places, since the question gives the sides this way.

00:15 And our angles by default will be given as final answers to the nearest minute.

00:23 So our first unknown will be the angle alpha.

00:31 To find this, we should find a trig ratio of alpha that consists of the two known sides, because this will allow us to get an equation where the only unknown is alpha, and we will solve for that.

00:43 So notice the known side a is opposite to alpha, and the known side b is adjacent to alpha.

00:53 A trig function that uses opposite and adjacent is tan.

00:56 And we can see tan of alpha would be equal to the opposite, 6 .00 over the adjacent 8 .46.

01:08 Finally, to isolate alpha, we will apply inverse tan to both sides.

01:13 To get that alpha is equal to inverse tan of 6 .00 over 8 .46.

01:23 When entered into the calculator, we get alpha is equal to 35 .345 degrees.

01:35 Approximately and we round to three decimal places in this case because this allows us to accurately calculate the amount of minutes so we will convert this to degrees and minutes by rewriting this as 35 degrees plus 0 .345 degrees and degrees and this 0 .35 degrees can be converted to minutes by multiplying it by 60 since there are 60 minutes in a degree.

02:14 So when we multiply that by 60 to the nearest minute, that second part will become 21 minutes.

02:31 Therefore, our final answer for alpha is 35 degrees and 21 minutes.

02:39 Our next unknown is angle beta.

02:42 Now, we could do the same process.

02:47 However, it would be easier to just use the angle sum and the angles that we now have, because the angle sum of any triangle is 180 degrees, and we now have the other two angle measures.

03:02 So we can say the angle sum, beta, plus the right angle, 90 degrees, and zero minutes, plus the angle alpha or 35 degrees and 21 minutes is equal to 180 degrees...

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