【Math I】Mastering Figures and Measurement (Trigonometric Ratios)!
Hello everyone! In this chapter, we will be studying "Figures and Measurement." The name might sound a bit intimidating, but it is essentially the field that investigates the "relationship between the angles and side lengths of a triangle."
You will acquire "magical tools" that allow you to calculate things like the height of the Tokyo Skytree, which cannot be measured directly, simply by knowing the angles of a triangle. At first, the calculations might feel a bit overwhelming, but once you grasp the basic rules, you'll be just fine. Let’s take it one step at a time together!
1. Basics of Trigonometric Ratios (Sine, Cosine, Tangent)
Let's start by defining trigonometric ratios using right-angled triangles. Focusing on the angle \(\theta\) (theta), let's memorize these three names.
If we denote the hypotenuse of the right triangle as \(r\), the base as \(x\), and the height as \(y\), they are defined as follows:
- Sine (\(\sin \theta\)): \(\frac{y}{r}\) (Height divided by hypotenuse)
- Cosine (\(\cos \theta\)): \(\frac{x}{r}\) (Base divided by hypotenuse)
- Tangent (\(\tan \theta\)): \(\frac{y}{x}\) (Height divided by base)
【Tips for Memorization!】
A famous method is to imagine drawing the cursive letters of the alphabet over the triangle!
・sine (s): Trace from the hypotenuse to the height.
・cosine (c): Trace from the hypotenuse to wrap around the base.
・tangent (t): Trace from the base to the height.
Key Point: Trigonometric Ratios of Special Angles
You absolutely must know the values for 30°, 45°, and 60° for your tests! Rather than just memorizing them, it's more powerful to recall the "ratios of your set squares" and calculate them on the spot.
・The ratio of sides for a 30° right triangle is \(1 : \sqrt{3} : 2\)
・The ratio of sides for a 45° right triangle is \(1 : 1 : \sqrt{2}\)
【Summary】 Trigonometric ratios are simply "ratios of side lengths." Start by drawing a right triangle and thinking it through!
2. Relationships Between Trigonometric Ratios
Sine, cosine, and tangent are not separate entities; they are actually deeply connected to each other. The following three formulas are super important:
- \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
- \(\sin^2 \theta + \cos^2 \theta = 1\)
- \(1 + \tan^2 \theta = \frac{1}{\cos^2 \theta}\)
【Common Mistake】
The notation \(\sin^2 \theta\) means "the square of the entire \(\sin \theta\)." Be careful not to write \(\sin \theta^2\), which looks like you are squaring the angle \(\theta\) instead!
【Fun Fact】
The second formula, \(\sin^2 \theta + \cos^2 \theta = 1\), is actually the same thing as the "Pythagorean Theorem" you learned in junior high. Doesn't that make it feel a bit more familiar?
【Summary】 Once you know one trigonometric ratio (for example, just \(\sin \theta\)), you can use these formulas to calculate the other two!
3. Extended Trigonometric Ratios (0° to 180°)
Until now, we have only thought about right triangles (up to 90°), but from here on, we will handle obtuse angles (angles greater than 90°).
Here is where the "unit circle" (a circle with a radius of 1) comes in.
- \(\sin \theta\) is the \(y\)-coordinate
- \(\cos \theta\) is the \(x\)-coordinate
- \(\tan \theta\) is the slope of the line
Remember it this way! This allows you to find values for 0° and 180°, where you can't even form a triangle.
【Important Points!】
・\(\sin \theta\) is always positive in the range from 0° to 180° (imagine a hill).
・\(\cos \theta\) becomes negative once it exceeds 90°. This is a common place to make mistakes!
【Summary】 If the angle exceeds 90°, use the unit circle. Don't forget that cosine becomes negative!
4. The Law of Sines and the Law of Cosines (Triangle Calculations)
It's time for the real deal. Here are two powerful weapons that can be used on any triangle (not just right triangles!).
① The Law of Sines
Formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R\)
(Note: \(R\) is the radius of the circumcircle of the triangle.)
When to use it?
・When you know a "pair of opposite side and angle."
・When the problem mentions the "radius of the circumcircle."
② The Law of Cosines
Formula: \(a^2 = b^2 + c^2 - 2bc \cos A\)
When to use it?
・When you know "two sides and the angle between them."
・When you know the "lengths of all three sides" and want to find an angle.
【Tips for Memorization】
It's easy to remember the Law of Cosines if you think of it as the Pythagorean Theorem (\(a^2 = b^2 + c^2\)) with an extra term added on!
【Summary】 For "opposite pairs," use the Law of Sines; for "two sides and the included angle," use the Law of Cosines. Choose your weapon based on the situation!
5. Area of a Triangle
You can find the area using methods other than base \(\times\) height \(\div 2\). By using trigonometric ratios, you don't even need to know the height!
Formula: \(S = \frac{1}{2} bc \sin A\)
Memorize it in words:
"Area = \(\frac{1}{2} \times\) the product of two adjacent sides \(\times\) the sine of the angle between them."
【Common Mistake】
The formula for area uses \(\sin\)! It's easy to get it confused with the Law of Cosines, so remember the pair: "Area uses sine."
【Summary】 If you have two sides and the \(\sin\) of the angle between them, you can find the area instantly!
Finally
The field of figures and measurement might seem difficult at first because of all the formulas. However, it’s actually like solving a puzzle—it’s all about knowing "which formula to use and when."
Start with simple problems, practice writing the formulas down on paper, and get used to the calculations. It’s okay to take it slowly at first. As you solve more problems, you’ll naturally know how to choose the right one! I’m rooting for you!