Here is a brief chat I had asking about trig. If you haven't gone through this exercise, please read it. I'm of the opinion that this is going to be immensely useful. This is much better than a typical textbook, if for no other reason than you don't have to find the information yourself. The language is clear and concise, and you can ask for clarification on specific things without needing to look for it elsewhere. I self taught myself a few AP courses with a book. I feel like I would have been MUCH MUCH BETTER OFF if I could have used this.

ME: what's trigonometry

GPTPlus:
Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It is used to solve problems involving triangles and is also applied in various fields such as physics, engineering, and astronomy. Trigonometric functions such as sine, cosine, tangent, cosecant, secant, and cotangent are used to calculate the values of angles and sides of triangles. These functions have important applications in fields such as navigation, architecture, and computer graphics. Trigonometry also has connections with complex numbers and calculus, making it a fundamental subject in mathematics.

Me: I need to use sine, cosine, and tangent. What are those. how do they work

GPTPlus:
Sine, cosine, and tangent are the three most common trigonometric functions used in mathematics. They are abbreviated as sin, cos, and tan, respectively. Each of these functions relates the angle of a right triangle to the ratio of two of its sides.

Here's a brief explanation of each of these functions:

Sine (sin): The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In other words, sinθ = opposite/hypotenuse.

Cosine (cos): The cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse. In other words, cosθ = adjacent/hypotenuse.

Tangent (tan): The tangent of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the adjacent side. In other words, tanθ = opposite/adjacent.

To use these functions, you need to know the values of at least two sides of a right triangle and the angle of interest. You can then use the appropriate trigonometric function to find the value of the unknown side or angle.

For example, if you know the length of the adjacent side and the angle, you can use the cosine function to find the length of the hypotenuse. Or, if you know the length of the opposite side and the adjacent side, you can use the tangent function to find the angle.

________________________

Me: can you go through an typical example of how to use sin with actual numbers

GPTPlus:
Sure, I'd be happy to go through an example of how to use the sine function with actual numbers.

Let's say we have a right triangle with an angle of 30 degrees and a hypotenuse of 10 units. We want to find the length of the side opposite the 30 degree angle.

To use the sine function, we'll use the formula sinθ = opposite/hypotenuse, where θ is the angle of interest.

In this case, we have:

sin(30) = opposite/10

To solve for opposite, we can rearrange the formula:

opposite = sin(30) x 10

Now, we need to use a calculator or reference table to find the value of sin(30). The sine of 30 degrees is 0.5. So, we have:

opposite = 0.5 x 10

opposite = 5

Therefore, the length of the side opposite the 30 degree angle is 5 units.

This is just one example of how to use the sine function. Depending on the information given, you may need to use different trigonometric functions or formulas to solve for the unknown side or angle.