Trigonometry

Relating trig functions through angle rotations

$$角度\thetaのベクトルを反時計回りに\frac{\pi}{2}回転する$$ $$２つの単位円との交点の関係を表すと$$ $$\cos(\theta)=\sin(\frac{\pi}{2}+\theta)$$ $$\sin(\theta)=-\cos(\frac{\pi}{2}+\theta)$$ $$\tan(\theta)=-\frac{1}{\tan(\frac{\pi}{2}+\theta)}$$ $$\quad \tan(\frac{\pi}{2}+\theta) =\frac{\sin(\frac{\pi}{2}+\theta)}{\cos(\frac{\pi}{2}+\theta)}$$ $$\qquad =\frac{\cos(\theta)}{-\sin(\theta)} =-\frac{1}{\tan(\theta)}$$

Pythagorean identity

Pythagorean trig identity from unit circle

$$x=\cos\theta \quad y=\sin\theta$$ $$x^2 + y^2 = 1$$ $$\sin^2\theta + \cos^2\theta = 1$$

Angle addition formula for sine

$$\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$$
$$\sin(x+y) = \overline{DF} =\overline{DE}+\overline{EF} =\overline{DE}+\overline{CB}$$ $$\overline{EC} \parallel \overline{AB}$$ $$\angle ECA = y, \quad 90^{\circ}-\angle DCE = \angle CDE = y$$ $$\overline{AC}=\cos(x)$$ $$\sin(y)=\frac{\overline{CB}}{\overline{AC}} =\frac{\overline{CB}}{\cos{x}} \to \overline{CB}= \cos(x)\sin(y)$$ $$\overline{DC}=\sin(x)$$ $$\cos(y)=\frac{\overline{DE}}{\overline{DC}} =\frac{\overline{DE}}{\sin(x)} \to \overline{DE}= \sin(x)\cos(y)$$

Angle addition formula for cosine

$$\cos(x+y) = \cos(x)\cos(y) - \sin(x)\sin(y)$$
$$\cos(x+y) = \overline{AF} = \overline{AB}-\overline{FB} =\overline{AB}-\overline{EC}$$ $$\cos(y)=\frac{\overline{AB}}{\overline{AC}} =\frac{\overline{AB}}{\cos(x)} \to \overline{AB}=\cos(x)\cos(y)$$ $$\sin(y)=\frac{\overline{EC}}{\overline{DC}} =\frac{\overline{EC}}{\sin(x)} \to \overline{EC}=\sin(x)\sin(y)$$

Law of cosines and law of sines

Law of cosines

$$a^2=b^2+c^2-2bc\cos\theta$$

$$a^2=12^2+9^2-2(12)(9)\cos87^{\circ}$$ $$\quad = 225-216\cos87^{\circ}$$ $$a=\sqrt{225-216\cos87^{\circ}}\approx14.6$$

Law of sines

$$\frac{\sin30^{\circ}}{2} =\frac{\sin105^{\circ}}{a} =\frac{\sin45^{\circ}}{b}$$
が成り立つ $$\frac{\sin30^{\circ}}{2}=\frac{\frac{1}{2}}{2}=\frac{1}{4}$$ なので $$\frac{\sin105^{\circ}}{a}=\frac{1}{4} \to a=4 \cdot \sin105^{\circ}\approx 3.86$$ $$\frac{\sin45^{\circ}}{b}=\frac{1}{4} \to b=4 \cdot \sin45^{\circ} \approx1.83$$

Proof : Law of cosines

$$a^2=b^2+c^2-2bc\cos\theta$$

$$\cos \theta = \frac{d}{b} \to d=b\cos \theta$$ $$e = c - d = c - b\cos \theta$$ $$\sin \theta = \frac{m}{b} \to m=b\sin \theta$$ $$a^2 = m^2 + e^2$$ $$\quad = (b\sin\theta)^2 + (c - b\cos \theta)^2$$ $$\quad = b^2\sin^2\theta + c^2 - 2bc\cos\theta + b^2\cos^2\theta$$ $$\quad = b^2(\sin^2\theta + \cos^2\theta) + c^2 - 2bc\cos\theta$$ $$\sin^2\theta + \cos^2\theta=1なので$$ $$\underline{a^2 = b^2 + c^2 - 2bc\cos\theta}$$

Proof : Law of sines

$$\frac{\sin\alpha}{A}=\frac{\sin\beta}{B}$$

$$\sin\alpha = \frac{x}{B} \to B\sin\alpha = x$$ $$\sin\beta = \frac{x}{A} \to A\sin\beta = x$$ $$B\sin\alpha = A\sin\beta$$ $$\underline{\frac{\sin\alpha}{A} = \frac{\sin\beta}{B}}$$