Solve each problem. See Examples 1-4. Diameter of the Sun To determine the diameter of the sun,

Solve each problem. See Examples 1-4. Diameter of the Sun To...

Key Concepts

Trigonometry in Astronomical Measurements

Trigonometry is a key tool in astronomical observations and calculations since direct measurement of vast distances and sizes is often impractical. By using angular measurements and applying trigonometric formulas, astronomers can derive the dimensions of astronomical objects, such as the diameter of the Sun, based on known distances and measured angles.

Angle Measurement Conversion

This concept involves converting angles from one unit to another, such as from arcminutes or degrees into radians. In many trigonometric calculations, especially those involving circles and arcs, it is essential to work in radians because standard mathematical formulas, including those for sine and cosine, assume angles are measured in radians.

Chord Length in Circle Geometry

When an angle is subtended at the center of a circle, the straight-line distance (or chord) between the two points on the circle can be determined using the formula that relates the chord to the radius and sine of half the subtended angle. This relationship is foundational in problems involving circular arcs and is often used to approximate distances across large circular bodies.

Small-Angle Approximation

In scenarios where the angle subtended is very small, the sine of the angle (or half the angle) is nearly equal to the angle itself when measured in radians. This approximation greatly simplifies calculations in problems involving distant objects where the angles are typically small, allowing the chord length or diameter to be approximated directly as the product of the distance and the angle.

Transcript

00:01 Now, this question looks at right triangles and the applications of right triangles and seeing if you can decipher a word problem, really, and getting that measurement from the degrees, from the minutes or seconds to degrees, and then finding your angle that you're looking for.

00:25 So let's go ahead and look at an example here.

00:29 So you have this terrible looking mountain over here and you want to measure how wide this mountain is.

00:40 So let's say you're standing way out over here.

00:43 This is going to be small little us over here.

00:48 And we are looking at this mountain from super far away.

00:52 So let's say that we see this distance here, right? so we look at both ends here.

01:01 Let's say through binoculars or something, right? and we want to measure this distance, this vertical distance here, right? so from here to here, we want to know what that distance is.

01:14 And all we know is this distance, actually, we know how far we are from the mountain.

01:24 And we know how far we are, and we also have measured, right, looking at the binoculars, we say, okay, yes, this is going to be some degrees over here, and this is going to be some degrees over here, be some degrees over here.

01:34 So we know that the difference, the angle between them, this angle here, is going to be, let's say, 50 minutes.

01:48 Now, the conversion for 50 minutes to degrees is just 50 minutes over 60.

01:57 So the conversion from minutes to degrees is always 60.

02:00 You just divide by 60 to convert this to degrees here.

02:04 So let's see what our calculator tells us.

02:10 So our calculator, we're our calculators.

02:11 Says that this is going to be 0 .3 degrees right so that is what our degrees is here and we're going to call this angle here theta and what's convenient about this is if you look at it it's actually two right triangles right so there's going to be one right triangle here and there's actually another right triangle here right next to it so what we can do is we can actually divide data right so data here is this whole angle here but what we can do is we're actually going to take this small piece here which is half right because there's two of them so this is just half so we're actually going to divide this by two i'm actually going to divide this whole thing by two here and that gives us our new angle for our right triangle right so we want this angle because it's in the right triangle so if this the bigger triangle is harder to deal with so we want the small right triangle because right triangles have very nice properties that we know about so let's go ahead and divide this and that tells us that our angle that we should be concerned about is actually 4167, let's say degrees.

03:44 And now let's look at this distance here.

03:47 So let's say that this distance from you to the mountain is going to be, let's make it a nice number, let's say 100 ,000.

03:57 So we are super far from this mountain, right? so this is going to be 100 ,000 miles away.

04:06 Now, we have this side here, this one all the way up to the mountain, and we have this angle.

04:13 And what we want to find is this side here.

04:16 Let me just highlight this, this side here...

Please give Ace some feedback