Whether it is how far, how tall, or at what angle, knowing how to apply right angle trigonometry can help you to answer these questions.
Whether it is how far, how tall, or at what angle, knowing how to apply right angle trigonometry can help you to answer these questions. We start with a review of some basics about right triangles (0:45). If you know two sides of a triangle, then you can find the third side using the Pythagorean Theorem (1:10). If you know one side of a triangle and an angle in addition to the 90 degree angle, then you can find the remaining sides (2:10). What if you know two legs of a triangle - then you can use inverse trigonometric functions to find a missing angle (5:00).
The Question of the Day asks:(7:14)
Your mailbox is 10 m from your house, and the ledge of your bedroom window is located 5 m above the ground. How long would a piece of rope need to be to make a little zipline for your mail to travel to the box?
Thank you for listening to The APsolute RecAP: Physics 1 Edition!
(AP is a registered trademark of the College Board and is not affiliated with The APsolute RecAP. Copyright 2020 - The APsolute RecAP, LLC. All rights reserved.)
Website:
EMAIL:
Follow Us:
Hi and welcome to the APsolute Recap: Physics 1 Edition. Today’s episode will focus on the trigonometric functions.
Lets Zoom out:
This episode is going to be devoted to a mathematical skill that will be influential throughout numerous units in AP Physics. This is new to some students and review for others. Therefore, it is worth spending an episode reviewing some basics of trigonometry.
Lets Zoom in:
All the trigonometry we will do is specifically “right angle trig.” This means it will involve visualizing right triangles that have one angle measuring 90 degrees. As a reminder, the three angles of a triangle will sum to 180 degrees total, and the longest side is known as the hypotenuse while the other two sides are referred to as legs.
The first thing we will look at is known as the Pythagorean Theorem which states A squared plus B squared is equal to C squared. Or… in words, the length of the hypotenuse squared is equal to the sum of each leg squared. For example, if one leg is 3 meters long and the other leg is 4 meters long, then the hypotenuse is 5 meters long because 3 squared is 9, 4 squared is 16 and the sum of those values is 25 which is 5 squared. Not too bad at all!
So, what if the hypotenuse and one leg is known you ask? Like, if the hypotenuse is 10 m and one leg is 8 m? Well, then the other leg must be… … … 6 m, right. In this case we would square the hypotenuse 10 m to get 100 and then subtract 64 or 8 squared to get 36 which is 6 squared. Just as a reminder, you need to take the square root of the difference of 100 and 64 to find the measurement of a leg.
You can also work with the measurements of angles. For this, you will have to make sure you are in degree mode on your calculator. Usually, this involves pressing the MODE button and then highlighting the word degree rather than radian. We will work in radians later in the year, but for now let’s stick with degrees.
A popular question in the math realm is that you are located 200 meters from a redwood tree and you have to look up at an angle of 30 degrees to look at the top of the tree. We can use trig functions to determine the height of the tree with good ole’ SOH CAH TOA. More formally sine, cosine, and tangent. The sine of an angle is equal to the ratio of the opposite side over the hypotenuse hence SOH, “S” “O” “H”. The cosine of an angle is equal to the ratio of the leg adjacent or next-to the angle divided by the hypotenuse hence CAH, “C” “A” “H”. Lastly, we have tangent which is the ratio of… you guessed it the opposite leg to the adjacent leg which is why we have TOA, “T” “O” “A”. It can be hard for students new to using trig functions to know whether a side of a triangle is the adjacent leg or the hypotenuse. It is important to remember that the hypotenuse is the longest side and is opposite the 90 degree angle.
In the proposed problem regarding the redwood tree, tangent would be the trig function we would want to use since the hypotenuse is unknown and the leg adjacent to the 30 degree angle is 200 m in length, therefore the height of the tree is the leg opposite the 30 degree angle. And with some algebra, PRESTO CHANGEO, you arrive at a height of 115 m. (in a fast end a of a commercial legal style “All algebra used during the making of this podcast makes use of finite mathematics and does not in any way involve magic. All references to magic are for comedic purposes only and in no way reflect the exactness of mathematics”)
The other very helpful feature of using the trig functions is to determine an unknown angle. For example, a submarine that drives forward 1000 m while diving to a depth of 240 m would dive at an angle below the surface of the water. That angle can be found using an inverse trig function. Imagine drawing a horizontal line that represent the 1000 m drive forward and then making a 90 degree turn downward for 240 m. Connecting your starting and ending point with a line that is the hypotenuse. We know that the tangent of the angle we are looking for is equal to the ratio of the opposite leg (240m) over the adjacent leg (1000m). That ratio is equal to 0.24 and if we take the inverse tangent of 0.24 we will have our descent angle of 13.5 degrees. Just a reminder, you have to press 2nd on your calculator and then the TAN button in order to complete the inverse tangent function.
Probably the hardest part about using the trig functions in application word problems is being able to visualize the triangle that is relevant. It helps to sketch a picture.
One last thing! Most people refer to unknown angles as Theta, but truthfully you can use whatever letter you want from whatever alphabet since it is just an unknown variable. On the AP Physics 1 exam, you will often see it as theta, or a zero with a small horizontal line the middle of it. But, sometimes Phi or a zero with a small vertical line is used. It is absolutely worth your time and energy to practice using the trigonometric functions so because they will be a vital part of the course. I always like to look at it from that angle anyway… see what I did there?
To recap……
The Pythagorean theorem can be used to solve for a missing leg or the hypotenuse of a triangle. The trigonometric functions (sine, cosine, and tangent) can be used to solve for unknown legs of a triangle, the hypotenuse or a missing angle. An easy way to remember which ratios are equal to each trig function is SOH CAH TOA.
Coming up next on the APsolute RecAP Physics 1 Edition, we will be using the trigonometry covered in this episode and see how they apply to vectors of displacement, velocity, and acceleration.
Today’s Question of the Day focuses on applying the trigonometric functions.
Question: Your mailbox is 10 m from your house, and the ledge of your bedroom window is located 5 m above the ground. How long would a piece of rope need to be to make a little zipline for your mail?