【Math I】 Figures and Measurement (Trigonometric Ratios) — A Bridge Between Math and Geometry
Hello everyone! In this chapter, we’ll be learning about "Figures and Measurement."
You may have already heard the terms "sine, cosine, and tangent."
They might seem a bit intimidating at first, but they are actually just "convenient tools for calculating the side lengths of a triangle using its angles."
Once you master this unit, you'll be able to calculate things you can't measure directly, like the height of a mountain or the length of a building's shadow. This is an essential topic that appears frequently on common exams, so let's take it one step at a time at your own pace!
1. Basics of Trigonometric Ratios (Starting with Right-Angled Triangles)
Let’s start with the basics: the definition using right-angled triangles. If we let one of the acute angles of a right-angled triangle be \( \theta \) (theta), the ratios of the three sides (trigonometric ratios) are fixed.
(1) Definitions of Sine, Cosine, and Tangent
① Sine
\( \sin \theta = \frac{\text{height}}{\text{hypotenuse}} \)
② Cosine
\( \cos \theta = \frac{\text{base}}{\text{hypotenuse}} \)
③ Tangent
\( \tan \theta = \frac{\text{height}}{\text{base}} \)
💡 Pro-Tip for Memorization (Cursive Imagery)
Try visualizing the cursive letters of the alphabet:
・sin: Trace from the hypotenuse to form a cursive "s".
・cos: Wrap around the hypotenuse to form a cursive "c".
・tan: Start from the base and go up to form a cursive "t".
Try tracing these shapes with your finger to help them stick!
(2) Trigonometric Ratios for Special Angles (30°, 45°, 60°)
These are used so frequently that memorizing them is best, but if you forget, you can easily derive them from the "ratios of your set square rulers."
・45°, 45°, 90° isosceles right triangle → Ratio is \( 1 : 1 : \sqrt{2} \)
・30°, 60°, 90° right triangle → Ratio is \( 1 : \sqrt{3} : 2 \)
【Key Points】
\( \sin 45^\circ = \frac{1}{\sqrt{2}} \)
\( \cos 60^\circ = \frac{1}{2} \)
\( \tan 30^\circ = \frac{1}{\sqrt{3}} \)
Always make it a habit to draw a quick sketch to double-check!
【Key Takeaway】
Trigonometric ratios rely on the property that "if the angle is the same, the ratio of the sides remains constant regardless of the triangle's size."
2. Extending Trigonometric Ratios (Into the World of 0° to 180°)
Right-angled triangles alone can't handle angles of 90° or greater (obtuse angles). To fix this, we use the "unit circle (a circle with radius 1)" to broaden our perspective.
(1) Trigonometric Ratios Using Coordinates
Consider a point \( P(x, y) \) on the circumference of a unit circle, where \( \theta \) is the angle between \( OP \) and the positive direction of the \( x \)-axis:
・\( \sin \theta = y \) (The \( y \)-coordinate is the sine!)
・\( \cos \theta = x \) (The \( x \)-coordinate is the cosine!)
・\( \tan \theta = \frac{y}{x} \) (The slope is the tangent!)
🌟 Fun Fact: Why "Cosine"?
The "co-" in cosine stands for "complementary." Interestingly, \( \cos \theta \) has the same value as the "sine of the angle obtained by subtracting \( \theta \) from 90°" (\( \sin(90^\circ - \theta) = \cos \theta \)). Pretty cool, right?
(2) A Note on Signs (Plus/Minus)
For obtuse angles (90° < \( \theta \) < 180°), the \( x \)-coordinate of the point becomes negative.
・\( \sin \theta \) remains positive up to 180° (because it's the \( y \)-coordinate).
・\( \cos \theta \) becomes negative past 90° (because it's the \( x \)-coordinate).
・\( \tan \theta \) becomes negative past 90° (because the slope tilts downward to the right).
【Common Mistake】
A very common error is keeping the cosine of an obtuse angle positive!
Always visualize that "for obtuse angles, \( \cos \) moves to the left (negative)."
3. Relationships Between Trigonometric Ratios (Crucial Formulas)
These three formulas are like puzzle pieces that work together. Be sure to memorize them!
① \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
② \( \sin^2 \theta + \cos^2 \theta = 1 \)
③ \( 1 + \tan^2 \theta = \frac{1}{\cos^2 \theta} \)
【Point】
If you know either \( \sin \theta \) or \( \cos \theta \), you can use formula ② to find the other. When taking the square root, always check the range of \( \theta \) to determine if it should be positive or negative.
4. The Sine Rule and Cosine Rule (Solving Triangles)
This is where things get exciting! Here are your powerful weapons that work for any triangle: the Sine Rule and the Cosine Rule.
(1) The Sine Rule
If \( R \) is the radius of the circumcircle of triangle \( ABC \):
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \)
When to use it?
・When you know "one pair of an opposite side and angle" plus "one other angle or side."
・If you see the term "circumradius \( R \)", you will use this 100% of the time!
(2) The Cosine Rule
\( a^2 = b^2 + c^2 - 2bc \cos A \)
(Rearranged form: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \))
When to use it?
・When you know "two sides and their included angle" and want to find the remaining side.
・When you know the lengths of "all three sides" and want to find an angle.
💡 Pro-Tip for Memorization
Doesn't the structure of the Cosine Rule look like the Pythagorean theorem (\( a^2 = b^2 + c^2 \))?
Just remember it's the Pythagorean theorem with a "little bonus" of \( -2bc \cos A \) at the end.
【Key Takeaway】
・If you see circumradius \( R \) → Use the Sine Rule!
・If you have 3 sides or 2 sides with an included angle → Use the Cosine Rule!
5. Geometric Measurement (Area and Volume)
Finally, let’s use trigonometric ratios to calculate areas smartly.
(1) Area of a Triangle \( S \)
\( S = \frac{1}{2} bc \sin A \)
You can find the area using "\( \frac{1}{2} \times \) side \(\times\) side \(\times\) sine of the included angle." No need to know the height!
(2) Inradius and Area
\( S = \frac{1}{2} r (a + b + c) \)
(where \( r \) is the radius of the incircle)
Using this, you can work backward from the triangle's area to find the inradius.
【Step-by-Step: Solving Complex Problems】
1. Start by drawing the given information on a diagram.
2. Define the value you want to find as \( x \).
3. Decide whether the Sine Rule or Cosine Rule is applicable.
4. Set up the equation and solve (watch out for calculation errors with square roots!).
Closing
Great job finishing this study session on "Figures and Measurement"!
It might feel like a lot of formulas at first, but by solving problems repeatedly, you’ll intuitively know which formula to use in which situation.
On the common exam, you'll be tested on your ability to "spot right-angled triangles within larger figures" and "determine which theorem is the most efficient to use."
Start by practicing the basic textbook problems to build your "formula usage" skills. You've got this—keep going one step at a time, and this will become one of your strongest subjects!