Trigonometry


Basics


角度の表し方 度数(degrees)と弧度(radians)

angledegreesradians
$$a$$$$360$$$$\quad2\pi$$
$$b$$$$180$$$$\quad\pi$$
$$c$$$$ 90$$$$\quad \frac{\pi}{2}$$
$$\quad 0$$$$\quad 0$$
$$d$$$$-90$$$$-\frac{\pi}{2}$$
弧度と度数の変換 $$1°=\frac{\pi}{180}$$ $$1_{(rad)}=\frac{180}{\pi}$$

$$radians = degrees \times \frac{\pi}{180} $$


三角関数(trigonometric function)

三角形ABCで
AC:hypotenuse
AB:adjacent
BC:opposite to the θ

sin cos tan
soh
sine is opposite over hypotenus
cah
cosine is adjacent over hypotenuse
toa
tangent is opposite over adjacent
$$sin\theta=\frac{opposite}{hypotenuse}$$ $$cos\theta=\frac{adjacent}{hypotenuse}$$ $$tan\theta=\frac{opposite}{adjecent}$$

Sine and cosine of complements

三角形ABCで
角Bは直角
角Cは58°
ならば $$cos58° \approx 0.53$$ このとき $$sin\theta=?$$

三角形の内角の和は180° → 角B=90°なので 角A + 角C = 90°

$$ \theta = 90° - \angle C = 90° - 58° = 32°$$

$$cos58°=\frac{BC}{AC}, \quad sin32°=\frac{BC}{AC}$$ $$\therefore$$ $$sin32°=cos58° \approx 0.53$$

$$sin\theta = cos(90°-\theta)$$ $$cos\theta = sin(90°-\theta)$$


Secant(sec), cosecant(csc) and cotangent(cot)

$$sin\theta$$ $$cos\theta$$ $$tan\theta$$
$$\frac{opp}{hyp}=\frac{12}{13}$$ $$\frac{adj}{hyp}=\frac{5}{13}$$ $$\frac{opp}{adj}=\frac{12}{5}$$
Reciprocal trig functions
$$csc\theta$$ $$sec\theta$$ $$cot\theta$$
$$\frac{hyp}{opp}=\frac{13}{12}$$ $$\frac{hyp}{adj}=\frac{13}{5}$$ $$\frac{adj}{opp}=\frac{5}{12}$$