Solve each right triangle. In Exercise 46, give angles to the nea...

Question

Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46.

A = 39.72°, b = 38.97 m

Accepted Answer

Hey, everyone here we are asked to find the missing side lengths and the missing angle of the right triangle. Where here we are given angle C is degrees angle A is 32.54 degrees and side length B is 24. centimeters. Here we have four answer choice options, answer choices A through D, each of which state A varying angle for angle B and lengths for side links A and C. So to begin this problem, we can start by finding our missing angle where here we are missing angle B and so to do this, we need to first recall the angle sum property of a triangle which states that angle A plus angle B plus angle C is equal to degrees. And now just substituting in our given information, we have 32. degrees plus unknown angle B plus 90 degrees is equal to degrees. And now we just want to solve four angle B and so that means we just subtract 90 degrees and 32.54 degrees from both sides of the equation. And we find that angle B is equivalent to 57.46 degrees. And now that we have found our missing angle, we can move on to finding our missing side links. And so beginning with, let's say the value of A or side length A, we need to recall the definition of a tangent which states that tangent A or tangent of angle A is equivalent to side length A divided by side length B. And now just substituting in the information we already have, we have tangent of 32.54 degrees is equal to unknown side length A divided by 24.75. And now solving for side length A, we multiply both sides of the equation by 24.75. In doing this, we have that side length A is equivalent to 24.75 multiplied by tangent of 32. degrees. And now just computing this expression, we find that side length A is equal to 15.79 centimeters. And now that we have found side length A, we can move on to our other missing side length which is side length C. And for side length C, we have to recall the definition of cosine, which states that cosine of angle A is equivalent to side length B divided by side length C. And now substituting in our given values, we have cosine of 32.54 degrees. Is equivalent to 24. divided by C. And now we just need to solve this equation for C. So first, we need to multiply both sides of the equation by C. And so we have C multiplied by cosine of 32.54 degrees is equal to 24.75. Continuing to solve for ac, we now need to divide both sides of the equation by cosine of 32.54 degrees. And again, we do this to both sides. And so we have that C is equal to 24.75 divided by cosine of 32. degrees. And now just computing this expression, we find that side length C is equivalent to 29.36 centimeters. And with this, we have found our missing side links A and C as well as our missing angle B. And so with these values, we are left with answer choice D where again, angle B is 57.46 degrees. Side length A is 15.79 centimeters and side length C is 29. centimeters. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each right triangle. In Exercise 46, give angles to the nearest minute. In Exercises 47 and 48, label the triangle ABC as in Exercises 45 and 46. A = 39.72°, b = 38.97 m

Key Concepts

Right Triangle Properties

Trigonometric Ratios

Angle Measurement

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