直角三角形公式和余函数（余角）公式

The three basic functions

In a Right-angled Triangle, take one of the angles that isn't the right angle and call it $x$. The side that is the longest and is opposite to the right angle is called the Hypotenuse, as you already know.

The side that is next to the angle $x$ and isn't the Hypotenuse is called the Adjacent side, a good way to remember this is that adjacent means "next to"

The remaining side is called the Opposite side, you can remember this because the side is opposite to the angle $x$

The basis of Trigonometry is dependent on the fact that a triangle is stable.

Try taking 3 toothpicks or pencils or other stright objects and put them into the shape of a triangle. Now try changing the shape of the triangle. You will realise you can't change it, because a Triangle is stable.

Since a triangle is stable, If you know the 3 angles in a triangle, the ratios of the sides will always be the same.

In trigonometry, the three main ratios used are sine, cosine, and tangent. Which is often abbreviated to $\sin$, $\cos$, and $\tan$.

For an angle $x$, this is how we define the three ratios:

$$\sin x = \frac{Opposite}{Hypotenuse}$$ $$\cos x = \frac{Adjacent}{Hypotenuse}$$ $$\tan x = \frac{Opposite}{Adjacent}$$

A good way to Memorise this is: "SOHCAHTOA", it stands for:

Sine is Opposite divided by Hypotenuse

Cosine is Adjacent divided by Hypotenuse

Tangent is Opposite divided be Adjacent

And we find that the Opposite side has length $2$.

A useful acronym:

SOHCAHTOA

This stands for

Sine: Opposite/Hypotenuse

Cosine: Adjacent/Hypotenuse

Tangent: Opposite/ Adjacent

Right triangles are one of the most important and influential topics in all of geometry and most of trigonometry can be based upon them. Trigonometry involves three basic functions, $\sin\theta, \cos\theta, \tan\theta$, where $\sin\theta$ stands for the sine of angle $\theta$, $\cos\theta$ stands for the cosine of angle $\theta$, and $\tan\theta$ stands for the tangent of angle $\theta$. These functions can be defined in many ways, but the first definition of them comes from right triangles. In a right triangle, you can take the sine of an angle x by dividing the side opposite the angle by the hypotenuse. Similarly, cosine of x can be found by dividing the side adjacent to the angle by the hypotenuse. Tangent is defined as the opposite over the adjacent sides. This gives us,

$$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$$ $$\cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$ $$\tan\theta = \frac{\text{opposite}}{\text{adjacent}}$$

In addition to these three functions, there are three more trigonometric functions used, called the reciprocal trigonometric functions. These are $\csc\theta, \sec\theta, \cot\theta$, which stand for cosecant, secant, and cotangent of $\theta$. These are just reciprocals of the original three functions, hence the name. They can be defined as,

$$\csc\theta = \frac{\text{hypotenuse}}{\text{opposite}}$$ $$\sec\theta = \frac{\text{hypotenuse}}{\text{adjacent}}$$ $$\tan\theta = \frac{\text{adjacent}}{\text{opposite}}$$

There are three types of classifications for triangles based on the size of the angle:

Obtuse: one angle > 90 degrees

Right: one angle = 90 degrees

Acute: one angle < 90 degrees

In the case of right triangles, it is possible to use basic trigonometric functions to find angles, side lengths, and therefore solve real-world problems involving trigonometry.

There are three basic trigonometric functions: Sine, Cosine, and Tangent.

There is a simple acronym that can help you remember the definition of each function

S-O-H-C-A-H-T-O-A.

SOH stands for (Sine=Opposite/Hypotenuse). This is the ratio that defines the Sine value for a certain angle. It is the length of the side opposite the considered angle divided by the length of the hypotenuse of the triangle.

CAH stands for (Cosine=Adjacent/Hypotenuse). This is the ratio that defines the Cosine value for a certain angle. It represents the length of the side adjacent to the considered angle divided by the length of the hypotenuse of the triangle.

Finally, TOA stands for (Tangent=Opposite/Adjacent). It represents, well, this time you get to answer a question about it!

This is because the Pythagorean theorem tells us that, if we have a triangle with legs* of lengths $a,b$ and hypotenuse (side opposite right angle) of length $c$, the hypotenuse is $c = \sqrt {a^2 + b^2}$.

• Remember that the legs are the side of the right triangle which intersect at a right angle (i.e. are perpendicular to each other).

Since $c^2 = a^2 + b^2$ and $a^2, b^2, c^2 \ge 0$, we know that $c^2 \ge a^2, b^2$, so $c \ge a, b$. Hence, the hypotenuse (the side opposite the right angle) must be the longest.

Incidentally, we can make a stronger statement and say $c > a, b$ because if $a$ or $b$ is equal to $0$, it's not a triangle anymore (it's a segment, which is also a special case of a degenerate triangle).

Finally, since $AB$ is the longest side, it must be the hypotenuse, so the angle opposite $AB$, namely $\angle ACB$ is the right angle.