Depth resolution: Why use unit circle learning triangles?
In junior high mathematics, we first contact the triangle function by " straight angle triangle": the siding is the sine, the siding is the cosine. But there's a fatal flaw in this definition -- it can only explain. 0° to 90° A point between. When we enter high school and university and get to the blunt, negative and even infinite angles, the definition of the straight-angle triangle is invalidated. This time, Unit Circle It becomes the geometric model that understands the core of the trigonometric function.
What's a circle?
In the square arcs, with the original point $ (0,0) for the centre, radius $r = $1 The circle is called the unit circle. On a unit circle, any one of the angles from its original point of origin with the positive half-axis of $x$$\theta$ , and the rays, with a unit circle at a point of $P(x,y). By definition:
- The cross-coordinate of the dot is the cosine of that angle: $\cos(\theta) = x$
- The vertical coordinates of the dot are the sines of that angle: $\sin (\theta) = y$
- The slope of the line is the tangent: $\tan(\theta)=\frac{y}$
You can drag any point on a circle with the interactive presentation tool above. You can see in a visual way: however the angle changes (or even exceeds) 360°), $\sin$ and $\cos' values are always limited between $[1, 1], precisely because the radius of the circle is 1.
Core formula derived from the circle
Using the geometry of the numeric theorem and the unit circle, we can easily extrapolate the most important equations in high school mathematics:
sin²(θ) + cos²(θ) = 1² = 1
tan(θ) = sin(θ) / cos(θ)
About y-axis symmetry: sin = sin
For origin symmetry: sin (π) = -sin (θ)
radians and angles
In this tool you can also see the conversion of angles (Degree) and radians (Radian). In calculus and higher mathematics, we usually impose the radians. Why?
Because the definition of radians is: Arc length divided by radius 。 In a circle (radio 1) the radians are actually equal to the corresponding " arc length " . Round's perimeter is $2\pi r = So a radians circle (360°) corresponds to $2pi. This makes us do triangulations. The formula is the most concise when $\frac{d}\sin x=\cos x$). Conversion formula:
1 radians = (180/gill) ° ≈ 57.3°
radium (radians) = angle x (thorium/180)
How can this tool be used efficiently?
1. Image memory symbol pattern: The drag angle passes four radiums and observes a positive or negative change. First quadrant is straight, second quadrant is positive, third quadrant tan Zero, fourth quadrant cos. Zero. The mouth.
2. Understanding the tangent: Watch the yellow tan line. When the angle is close to 90°, tan The line becomes infinite. This is because the line is now closer to vertical and the slope is closer to infinite.
3. Table of reference: I've met a job like $120 ^ Circ. , and in addition to the calculator, you can obtain a precise expression with root numbers by clicking on the shortcut button above or by looking at the table of special angles at the bottom (e.g. $\frac{\sqrt{3}}{2}$）。