In three-dimensional space, points are represented by ordered triples $(x, y, z)$. The distance formula in 3D is an extension of the 2D Pythagorean theorem:
For two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the distance $d$ between them is:
$$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}$$
The midpoint formula in 3D is similarly extended:
$$M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2})$$
Example
Find the distance and midpoint between $A(1, 2, 3)$ and $B(4, 6, 8)$.
Distance: $d = \sqrt{(4-1)^2 + (6-2)^2 + (8-3)^2} = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{50} = 5\sqrt{2}$
Midpoint: $M = (\frac{1+4}{2}, \frac{2+6}{2}, \frac{3+8}{2}) = (2.5, 4, 5.5)$
Understanding the formulas for volume and surface area of various 3D solids is crucial. Here are some key formulas:
Note
Remember that for composite solids, you may need to break them down into simpler shapes and add or subtract their volumes/surface areas as appropriate.
In 3D geometry, understanding the relationships between lines and planes is crucial. Here are some key concepts:
Example
If two planes have normal vectors $\vec{n_1} = (1, 2, 3)$ and $\vec{n_2} = (2, -1, 2)$, the angle $\theta$ between them can be found using the dot product formula:
$$\cos \theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}||\vec{n_2}|}$$
$$\cos \theta = \frac{1(2) + 2(-1) + 3(2)}{\sqrt{1^2 + 2^2 + 3^2}\sqrt{2^2 + (-1)^2 + 2^2}}$$
$$\cos \theta = \frac{6}{\sqrt{14}\sqrt{9}} = \frac{6}{\sqrt{126}}$$
$$\theta = \arccos(\frac{6}{\sqrt{126}}) \approx 58.1°$$
In a right-angled triangle, the trigonometric ratios are defined as:
Tip
Remember the mnemonic SOH-CAH-TOA to recall these ratios easily.
These ratios are fundamental to solving problems involving right-angled triangles.
Common Mistake
Students often confuse which side is opposite or adjacent. Remember, the opposite side is always across from the angle in question, while the adjacent side is next to it (not including the hypotenuse).
For non-right angled triangles, the sine and cosine rules are essential tools:
Sine Rule: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Cosine Rule: $$c^2 = a^2 + b^2 - 2ab \cos C$$
Where $a$, $b$, and $c$ are the lengths of the sides opposite to angles $A$, $B$, and $C$ respectively.
Example
In a triangle ABC, if $a = 5$, $b = 7$, and $C = 60°$, find angle $A$:
Using the sine rule: $$\frac{5}{\sin A} = \frac{7}{\sin 60°}$$
$$\sin A = \frac{5 \sin 60°}{7} = \frac{5\sqrt{3}}{14}$$
$$A = \arcsin(\frac{5\sqrt{3}}{14}) \approx 37.8°$$
Trigonometry has numerous real-world applications:
Example
A ladder 10 meters long leans against a wall. If the bottom of the ladder is 2 meters from the wall, how high up the wall does the ladder reach?
This forms a right-angled triangle. Let $\theta$ be the angle between the ground and the ladder.
$$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{10}$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{2}{10} = 0.2$$
$$\theta = \arccos(0.2) \approx 78.46°$$
Now we can find the height: $$h = 10 \sin 78.46° \approx 9.8 \text{ meters}$$
Radians provide an alternative way to measure angles, often preferred in advanced mathematics due to their natural relationship with the unit circle.
Definition: One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.
Key conversions:
Note
The formula for converting between degrees and radians is: $$\text{radians} = \frac{\pi}{180} \times \text{degrees}$$ $$\text{degrees} = \frac{180}{\pi} \times \text{radians}$$
The unit circle is a circle with a radius of 1 centered at the origin (0,0). It's a powerful tool for understanding trigonometric functions.
On the unit circle:
Special angles and their exact values:
Angle $\sin \theta$ $\cos \theta$ $\tan \theta$ 0° 0 1 0 30° $\frac{1}{2}$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{3}}{3}$ 45° $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{2}}{2}$ 1 60° $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $\sqrt{3}$ 90° 1 0 undefined
Tip
Memorizing these special angles and their values is crucial for solving trigonometric problems efficiently.
Trigonometric identities are equations that are true for all values of the variables involved. Some fundamental identities include:
Example
Prove that $\tan^2 \theta + 1 = \sec^2 \theta$
Start with the left side: $$\tan^2 \theta + 1 = (\frac{\sin \theta}{\cos \theta})^2 + 1 = \frac{\sin^2 \theta}{\cos^2 \theta} + 1$$
Multiply both terms by $\cos^2 \theta$: $$\frac{\sin^2 \theta}{\cos^2 \theta} \cdot \cos^2 \theta + 1 \cdot \cos^2 \theta = \sin^2 \theta + \cos^2 \theta$$
Use the Pythagorean identity: $$\sin^2 \theta + \cos^2 \theta = 1$$
Divide both sides by $\cos^2 \theta$: $$\frac{1}{\cos^2 \theta} = \sec^2 \theta$$
Thus, $\tan^2 \theta + 1 = \sec^2 \theta$ is proven.
Understanding the graphs of trigonometric functions is crucial for visualizing their behavior:
Transformations of these functions follow the general form: $$y = A \sin(B(x - C)) + D$$ Where:
Example
Graph $y = 2\cos(\frac{1}{2}x + \frac{\pi}{4}) - 1$
Solving trigonometric equations often involves using identities, factoring, and understanding the periodic nature of trigonometric functions.
General steps:
Example
Solve $\sin x = \frac{1}{2}$ for $0 \leq x
< 2\pi$
Therefore, the solutions are $x = \frac{\pi}{6}$ and $x = \frac{5\pi}{6}$.
The reciprocal trigonometric functions are:
These functions have their own unique properties and graphs:
Inverse trigonometric functions, denoted with "arc" or $^{-1}$, allow us to find angles given a trigonometric ratio:
