1 the hypotenuse of a right triangle measures 2sqrt15...

1. The hypotenuse of a right triangle measures $2\sqrt{15}$ centimeters and its shorter leg measures $2\sqrt{6}$ centimeters. What is the measure of the larger acute angle of the triangle? Round your answer to the nearest tenth of a degree. Answer: 2. Corey spies a bald eagle in a tall tree. He estimates the height of the tree to be 80 feet and the angle of elevation to the bird from where he stands to be $68^\circ$. The leaves on the tree make it difficult for Corey to watch the bird, so he takes several steps away from the tree to get a better view. He now estimates his angle of elevation to be $41^\circ$. How many feet did Corey step back to gain a better view of the bird? Round your answer to the nearest hundredth of a foot. Answer: 3. What is the exact value of the trigonometric expression? $$\frac{\tan\left(-\frac{2\pi}{3}\right)}{\sin\left(\frac{7\pi}{4}\right)} - \sec(-\pi)$$ State your answer in simplified radical form. Answer:

Transcript

00:01 All right, so we've got four trig problems.

00:02 We've got to solve here.

00:03 So the first one, it says solve the triangle.

00:06 What that means is find all the missing sides and missing angles.

00:10 Now, in this right triangle, we do have some stuff.

00:15 So angle c, the measure of angle c, we know is 90 degrees.

00:20 They told us the measure of angle b is 23 .2 degrees.

00:26 And the angle a is at the top, so the bottom side, the side across from angle a is side little a.

00:33 We know that is 2 .8 meters.

00:38 All right.

00:38 So those are the three things we know.

00:39 I marked them in red.

00:41 So now, real quick, to find angle a, we know that all three angles, all three of these angles here add to 180 right here.

00:57 So to find angle a, we just take the 90 and the 23 .2, add them together, subtract that from 180.

01:03 And it should give us 66 .8 degrees.

01:10 And so that's the measure of angle a.

01:15 So there we go.

01:15 All three of those should add up to 180.

01:18 Okay.

01:19 Now let's find side b.

01:21 Side b is this side over here between a and c.

01:26 It's a cross from angle b.

01:27 It's a right triangle.

01:30 If i use the 23 .2 degree angle, across from it is b, that's the opposite side.

01:38 2 .8 is the adjacent side.

01:41 So i can do tangent of 23 .2 is going to equal b over 2 .8.

01:49 And if i just multiply by 2 .8 on both sides, that'll tell me what b is type it into your calculator.

01:59 Now remember, we're using degrees, so we want to make sure our calculator is in degrees.

02:07 Sometimes they go into radians and we get weird answers.

02:13 All right, so i calculated it.

02:16 I got 1 .2 0 -0 -0, a bunch of zeros.

02:19 So 1 .2 is what b equals.

02:23 I'll put it on the line here.

02:25 1 .2 meters.

02:27 And now to find side c, that's the hypotenuse.

02:32 Now that we know side b, we could use the pythagorean theorem, or 2 .8 is still the adjacent side.

02:39 We want to find the hypotenuse adjacent and hypotenuse is the cosine.

02:44 So i could just do the cosine of 23 .2 is equal to the adjacent side over the hypotenuse.

02:53 And if we do some cross multiplying and get c by itself, c is going to be 2 .8 divided by the cosine of 23 .2 .2.

03:05 And then we type that into our calculator and we get 3 .04, so 3 .0 meters.

03:21 And so i like making this little table.

03:23 It keeps all my stuff nice and organized and neat.

03:27 So there you go.

03:28 So everything in blue were the three things we had to find.

03:31 Everything in red was given to us.

03:34 Okay.

03:35 Now let's move on to the second problem.

03:38 The inverse sign of the sign of 3 pi over 4.

03:42 Okay, be careful because a lot of times, and i do this two in my classes sometimes, i'll say, oh, well, the inverse sign and the sign, they just undo each other, so the answer is 3 pi over 4.

03:54 And that's mostly true.

03:57 But you got to remember, when we take the inverse sign, for sign to be invertible, we have to limit the domain of our, of the sign of theta.

04:10 We have to limit it.

04:12 Otherwise, it's not what's called one to one.

04:15 It's going to have, it's not going to pass the horizontal line test, which means it doesn't have an inverse.

04:20 So the inverse sign is only, the range of it is only from negative pi over two to pi over two.

04:33 So our answer has to be between negative pi over two and pi over two.

04:40 So what we need is we need the reference angle of 3 pi over 4.

04:44 One of the nice things about radiance is the denominator tells me, oh, my reference angle is pi over 4.

04:54 So our answer is going to be pi over 4...

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