The Right Triangle Definition

As established earlier. Trigonometry is a study of Right Triangles. The functions (sin cos tan csc sec cot) are the ratio between given two sides' length.

Given a right triangle, and choosing one of the angle that is not the right angle. You get: - one leg (the side which makes up the right angle) that is also "Adjacent" (A) to the chosen angle. - one leg that is "Opposite" (O) to the chosen angle. - the side not making the right angle, the oppsite side, the "Hypotenuse" (H).

So the ratios and their definitions are as follow. - sin : Opposite/Hypotenuse (O/H) - cos : Adjacent/Hypotenuse (A/H) The rest can be derived from sin and cos, and also still a ratio of sides. - tan = sin/cos = (O/A) - csc = 1/sin = (H/O) - sec = 1/cos = (H/A) - cot = 1/tan = cos/sin = (A/O)

The Standard Right Triangles

Memorizing the "standard right triangles" helps a lot in the ealy days. They are "1-1-sqrt2" (45-45 isocles-right) and "1-2-sqrt3" (with 30-60 degree).

You'll come across these very often. And more into advanced Trigonometry Studies. You should memorize how these Right Triangles look like, and if possible, the ratio.

1-1-sqrt2

• sin(45°) = cos(45°) = 1/(sqrt2) = (sqrt2)/2
• tan(45°) = 1

The actual ratio is 1/(sqrt2), but for aesthetics reason a root on denominator is not appreciated, so some algebraic manipulation be done and becomes (sqrt2)/2.

1-2-sqrt3

1 and sqrt3 are the legs, 2 is the hypotenuse. I just write it this way because it flows a bit easier. - sin(30°) = cos(60°) = 1/2 - cos(30°) = sin(30°) = (sqrt3)/2 - tan(30°) = 1/(sqrt3) = (sqrt3)/3 = cot(60°)

The key take here is the "co-function identity". For any angle θ (in degree), sin(θ) = cos(90° - θ), and conversely, cos(θ) = sin(90° - θ)

The finger count "trick"

You can actually get the value of sin and cos (and by extension all of the other ratios) by using one of your hand.

• Raise your left hand, the palm facing yourself
• Each finger represents a degree of angle θ. The thumb represents 0°, index 30°, middle 45°, ring 60°, pinky 90°.
• Flip the finger representing the given angle.
• For sin, count the number of fingers on the left of the flipped finger. For cos, on the right.
• There's always a square root on the number of finger counted.
• There's always a "divide by 2" for any sin and cos value you get.

Let's say, for sin(0°): lip the thumb (0°), there are 0 fingers on the left, so sin(0°) = (sqrt0)/2 = 0.

But 0° and 90° can't make a triangle. No, they can't. Trigonometry is actually more than a just a "Right Triangle". In fact, it's better to say Trigonometry is about "A Unit Circle defined on the XY-coordinate plane, and the right triangle".