In Exercises 1-12 , solve the right triangle shown in the figure. Round lengths to two decimal p

In Exercises 1-12 , solve the right triangle shown in the...

Key Concepts

Right Triangle

A right triangle is a triangle that has one angle measuring 90°. The geometric properties of right triangles enable the use of specific relationships and formulas—such as trigonometric ratios and the Pythagorean theorem—to solve for unknown sides and angles.

Trigonometric Ratios

Trigonometric ratios, including sine, cosine, and tangent, are relationships that connect the angles of a right triangle with its side lengths. They are used to determine the measurements of unknown sides or angles when at least one angle and one side are known.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle for right triangles, stating that the sum of the squares of the two legs equals the square of the hypotenuse. This theorem facilitates the calculation of a side length when the other two side lengths are known.

Complementary Angles

In a right triangle, the two acute angles are complementary, meaning their measures add up to 90°. This fact is useful in determining the measure of one acute angle when the other is known.

Rounding and Precision

Rounding in calculations is crucial for reporting results with a standard degree of precision. It involves approximating numerical values to a specified number of decimal places or degrees, ensuring consistency and clarity in mathematical communication.

Transcript

00:01 So here's the triangle, and my task is going to be to solve this triangle.

00:14 They've given me just a little bit of information, which i'm going to write here.

00:21 I don't know a, little a either.

00:26 Do you know b.

00:26 Angle b is 23 .8 degrees.

00:33 Side length b is 40 .5c, because it's a right triangle.

00:39 I know that's a 90 degree angle, and little c.

00:42 We also don't know yet.

00:44 So i'd like to have the information over there like so.

00:48 Now put this info here, 23 .8 goes here, that's 23 .8, and side b is 40 .5.

01:14 So in order to do this, we guarantee a trig function.

01:18 We're going to need a trig function and the function that i would recommend is sine so sign of angle b which is 23 .8 is going to equal the opposite side so opposite this angle is 40 .5 divided by the hypotenuse which is side c so first thing i want to do is multiply by c on both sides because that'll cancel here much better to have the variable in the numerator than the denominator and then i'll divide by the sign of 23 .8 on both sides because that will cancel there.

02:31 So c equals what's left over here is 40 .5 which i'm going to put in a calculator 40 .5 divided by sign of 23 .8.

02:42 Make sure you're in degrees not radians and i ended up with c equals 100.

02:47 0 .36.

02:57 1.

03:00 Don't like that.

03:05 There we go, a little bit.

03:09 Oh, sorry.

03:17 It's 100.

03:21 36.

03:22 I'm going to two decimal places because that's a side length...