Geometry & Trigonometry

3D Points and Shapes

When you are learning about the points on a 2-dimensional plane, you learn how to find distances between those two points. One of the easiest ways to derive that formula is by constructing a triangle and using Pythagoras. In the same way it is possible to derive a very similar expression for distance between two points in a 3D space:

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$$


You have also learned how to find the midpoint between two points on a 2d plane: by adding the individual coordinates together and dividing the sum by two. You can find the midpoint between two points in 3D space in almost the exact same way:

$$\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right)$$


On an exam, you will likely be asked to find the volume or surface of different 3D objects. IB expects you to know the formulas for area and volume of various shapes:

Shape Volume Surface Area
Right Pyramid
 $$\frac{1}{3}b^2h$$ Add the area of the four triangles and base rectangle that make up the pyramid:

$$4 \times \frac{1}{2}bh + b^2$$
Right Cone
 $$\frac{1}{3}\pi r^2 h$$ Add the area of the curved surface of a cone (\(\pi r l\)) and the area of the base circle for the total surface area:

$$\pi r l + \pi r^2$$
Cylinder
 $$\pi r^2 h$$ Add the areas of the top and bottom circles, and the (curved) rectangle connecting them:

$$2\pi r h + 2\pi r^2$$
Sphere
 $$\frac{4}{3}\pi r^3$$ Multiply 4 by the area of a circle with the same radius as the sphere to get the surface area:

$$4\pi r^2$$

Basic Trigonometry

Radians (rad): a radian is a unit of angular measure, similar to degrees. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle. There are 2π radians in a full circle, hence, 2π radians are equal to 360 degrees. It is important to know how to convert between radian and degrees:

$$ \begin{aligned} \text{rad} &= \text{deg} \times \frac{\pi}{180} \\ \\ \text{deg} &= \text{rad} \times \frac{180}{\pi} \end{aligned} $$

Circle Formulas:

  • Area of a sector: \(\frac{1}{2}r^2\theta\)
  • Arc length: \(r\theta\)

Where \(r\) is the radius, and \(\theta\) is in radians.

Right-Angle Triangles:

  • Pythagorean Theorem: \(a^2 + b^2 = c^2\)
  • Trigonometric Ratios:
    • Sine: \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\)
    • Cosine: \(\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}\)
    • Tangent: \(\tan\theta = \frac{\text{opposite}}{\text{adjacent}}\)

You will also need to recall the sin, cos, and tan of 0 degrees, 30 degrees, 45 degrees, 60 degrees, and 90 degrees. Rather than memorizing each of the values, you can use the two special triangles:

Non-Right Triangles:

For non-right triangles, SOH CAH TOA doesn't work, hence you will need to use the sine and cosine laws:

  • Sine Law: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\)
  • Cosine Law: \(c^2 = a^2 + b^2 - 2ab\cos C\)

Use the sine law when you have two angles and one side (AAS or ASA) or when you have two sides and a non-included angle.

Use the cosine law when you have either three sides (SSS) or two sides and the included angle (SAS).



Ambiguous Case

The ambiguous case is when you are given two sides and an angle not between those sides in a triangle (SSA). This case is "ambiguous" because it can result in one triangle, two triangles, or no triangle at all.


You have to use the sine rule to solve a problem in this case. However, one needs to remember that \(\sin x = \sin(180^\circ - x)\), meaning that your answer for an angle is not just \(x\), but also \(180^\circ - x\). In other words, we might get two different possible angles as an answer and thus two different possible triangles that satisfy the information given.


Unit Circle

The unit circle is a circle with a radius of 1 drawn from the origin of a set of axes. The y-axis corresponds to sine and the x-axis to cosine; so at the coordinate \((0, 1)\) it can be said that \(\cos\theta = 0\) and \(\sin\theta = 1\), just like in the \(\sin x\) and \(\cos x\) graphs when plotted.


Using the unit circle, we can see that since \(\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}\), it will be a positive value in quadrants 1 and 2, while \(\tan\theta\) will be a positive value in quadrants 1 and 3, and \(\cos\theta\) will be positive in quadrants 1 and 4.


Transformations

Besides the transformations in the functions chapter, trigonometric functions have some transformations with their own particular names. For a trigonometric function, the vertical stretch on a graph is determined by its amplitude, the horizontal stretch by its period and an upward/downward shift by its axis of oscillation.


A trigonometric function, given by \(y = a \cdot \sin(bx + c) + d\), has:


  • Amplitude of $a$: this is half the distance between its maximum and minimum y-values
  • Period of $360^\circ \over b$ or $2\pi\over b$: this refers to the length of one complete cycle
  • Horizontal shift of $+c$ to the left: in degrees or radians
  • Vertical shift of $+d$ upwards: this can be found by adding the maximum and minimum values and dividing by 2.

While you are not expected to know the graph of a tan function, it is important you know the basic graph and values of a sin and cos function, as well as how they are affected by the aforementioned transformations.


Identities and Equations

In order to solve trigonometric equations, you will sometimes need to use the identities listed below:

  • \(\tan\theta = \frac{\sin\theta}{\cos\theta}\)
  • \(\sin^2\theta + \cos^2\theta = 1\)
  • \(\sin(2\theta) = 2\sin\theta\cos\theta\)
  • \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)