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Physics: High School18 chapters | 211 lessons

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Lesson Transcript

Instructor:
*Damien Howard*

Damien has a master's degree in physics and has taught physics lab to college students.

This lesson reviews some basic trigonometry that is essential for an introductory physics course. Here we'll focus on the various methods for finding side lengths and angle measurements of a right triangle.

When you're working a physics problem, you are often dealing with a complex situation involving people, places, and things interacting with one another. In order to try to make sense of these situations, you can draw diagrams to understand what is going on in a physics problem. Some of the first diagrams you'll create will be for motion in two dimensions, and here you'll notice that right triangles show up quite a lot. In order to solve problems like these you'll need to know some trigonometry. In fact, you'll find you need trigonometry not just for 2-dimensional motion problems, but for many places in physics.

One of the most basic and essential things you will do with trigonometry is find the various side lengths and angle measurements of a right triangle. You can go about this in multiple ways, but one of the first ways you would have learned in a trigonometry course is known as the **Pythagorean theorem**, which is defined as:

*a*^2 + *b*^2 = *c*^2

The variables, *a*, *b*, and *c* are the three side lengths of a right triangle, where *c* is always the side opposite the right angle in the triangle. The Pythagorean theorem allows you to find the third side length of a triangle as long as you know the other two.

What if you only know one side length of a triangle, though? Well, as long as you know one of the two angles other than the right angle in the triangle you can still find the lengths of all the triangle's sides. To do this you use the **trigonometric functions**. The most common three are known as sine, cosine, and tangent. Theta represents the angle you know.

- sin(theta) = length of the opposite leg / length of the hypotenuse
- cos(theta) = length of the adjacent leg / length of the hypotenuse
- tan(theta) = length of the opposite leg / length of the adjacent leg

The hypotenuse is the side we called *c* in the Pythagorean theorem, and is always the side of the triangle across from the right angle. As their names suggest, the adjacent and opposite sides are the sides of the triangle adjacent and opposite to the non-right angle (theta) you are using.

These are the three trigonometric functions with which most people are familiar, but you might not know there are actually three more called cosecant, secant, and cotangent. Each of these three new trigonometric functions is actually the reciprocal of one of the previous functions we went over, where the reciprocal of any non-zero number or function is one divided by that same number or function.

- csc(theta) = length of the hypotenuse / length of the opposite leg
- sec(theta) = length of the hypotenuse / length of the adjacent leg
- cot(theta) = length of the adjacent leg / length of the opposite leg

You've gone over a few methods of finding the side lengths of a right triangle, but what do you do if you need to find the angles inside instead? If you know at least two of the side lengths you can use the **inverse trigonometric functions** to do this. These functions are represented with the prefix arc or an exponent of negative one, and there's a corresponding inverse trigonometric function for each of the six normal trigonometric functions.

- arcsin(length of the opposite leg / length of the hypotenuse) = theta
- arccos(length of the adjacent leg / length of the hypotenuse) = theta
- arctan(length of the opposite leg / length of the adjacent leg) = theta
- arccsc(length of the hypotenuse / length of the opposite leg) = theta
- arcsec(length of the hypotenuse / length of the adjacent leg) = theta
- arccot(length of the adjacent leg / length of the opposite leg) = theta

Each of these inverse trigonometric functions gives you the same answer. So which one you use depends on which side lengths of the triangle you know.

In physics, you will often find knowing trigonometry is necessary for solving problems. One of the most basic things you can do with trigonometry in a physics course is find the various side lengths and angle measurements for a right triangle.

When you have two side lengths you can use the **Pythagorean theorem** to find the third side length. If we know one of the angles other than the right angle and a side length, we can use the **trigonometric functions** to find the other side lengths. Finally, if you know at least two side lengths and want to find the triangle's angle measurements you can use the **inverse trigonometric functions**.

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Physics: High School18 chapters | 211 lessons

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