Solve the right triangle shown in the figure. Round lengths to...

Question

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. b = 2, c = 7

Accepted Answer

Hello everybody. I hope you're doing all right. Today, today we're going to be looking at this question that states find the solution for the right triangle in the given figure, expressing angles to the nearest 10th of a degree and rounding lengths to two decimal places. Now, the right triangle that we have, we see that uppercase letters denote angles and lowercase letters denote sides with the right angle being R and then we have another angle at Q and P and the sides of the same letters. But lower case we have PQ and R, they give us Q side side Q which is nine units and side R which is 21 units. Now, the answer choices provided are A, we have every answer choice has two angles. So P and Q will be angles and the second P will be side length P. So we have P equals 64.6 degrees and angle Q equals 25.4 degrees and sine P equals 96.18. That's for letter A as for choice, B says that angle P equals 66.46 degrees. Angle Q equals 25.4 degrees and side P equals 13.24 units. A choice C says that angle P is 64.6 degrees. Angle cube is 25.4 degrees and side P equals 18.96 units. And finally, as choice D says that angle P equals 64. degrees. Angle Q equals 18.9 degrees and side P equals 25.46 degrees. Now for a question dealing with a right triangle where we have to solve for the side. One of the first things we know is the Pythagorean theorem which is used to solve softer sides of a right triangle. So the Pythagorean theorem states that A squared plus B squared equals C squared where A and B are just 22 of the legs of the triangle and C is the hypotenuse. So if we rewrite this, we'll have P squared plus Q squared. Those are the two legs equals R squared because R is hypos and we can plug in with the values that we know we'll have P squared because we don't have P, we're solving for P plus 81. Or before we do that, we'll have plus nine squared equals squared. Now, if we simplify the squares, we'll have P squared plus 81 equals 441. And if we subtract 81 from the right hand side, we'll have P squared equals 360. And then we take the square of both sides, we'll have that P is around 18.97. That is the estimate. And then just like that we've salt for side length pink. Now, the next thing that we can do is solve for angle P and we do this by using something called. So that is soh dash cah dash to A and this is a shortcut of three letter abbreviations used to solve for trigonometric functions. So the first letter of each abbreviation stands for the function itself. So S for sign C for cosine T for tangents, the O stands for opposite, the H stands for hypotenuse and the A stands for adjacent. And now when you look at a abbreviation, we have the first letter with meaning function. Let's focus on cosine. So let's focus on cosine of P, let's say, so we have cosine of P is equal to the second letter divided by the third letter. So we have cosine of P is equal to the second letter A which is adjacent divided by the third letter which is hypo and this will be equal to nine or, and let's first label with the letter. So we'll have Q divided by R which is equal to nine divided by 21. Now to software, the angle itself will take that we have that the angle P is equal to the inverse cosine of that same fraction nine divided by 21. And when we plug that into our calculator, we'll have that angle P is around 64. degrees. That would be the estimate. And now finally to solve for the final angle, solve for angle Q, the last angle is easier than the other angle because we have two of the other angles already. And then the triangle, a triangle is equal to 100 and 80 degrees. So we have the, 100 and 80 degrees is equal to 90 degrees, which is the right angle plus 64.6 degrees, which is the angle that we calculated for P plus Q. So we'll have, we, if we subtract 90 from both sides, we'll have that 90 degrees is equal to 64.6 degrees plus Q. And now if we subtract 64.6 from the left hand side, we'll have 90 minus 64.6 degrees equals Q or that Q is equal to 25.4 degrees. And that will be our final answer for this question. And all of these match up with answer choice C, we see that the side length P is not exact. They have 18.96 and we have 18.97. But it's a close enough estimate. At the end of the day, we are estimating things here. But the other two answer choices do match up perfectly. So answer choice C is the best answer for this question. So we see how important it is to be able to know things like the Python theorem shortcuts like so and just know how a right triangle and a triangle in general functions with adding on to 100 and 80 degrees and such. So I hope that was helpful. And until next time.

Solve the right triangle shown in the figure. Round lengths to two decimal places and express angles to the nearest tenth of a degree. b = 2, c = 7

Key Concepts

Right Triangle Properties

Pythagorean Theorem

Trigonometric Ratios (Sine, Cosine, Tangent)

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