# TRIGONOMETRIA

• p.433 del libro ## Shapes We know the following volumes and surface areas:
• Volume of a cuboid: $$V_c = lwh$$
• Volume of right rectangular pyramid: $$Vp = \frac{lwh}{3} = \frac{1}{3}A{Base}h$$
• Area of circle: $$V_c = \pi r2$$
• Volume of cylinder: $$V_c = \pi r2h$$
• Volume of cone: $$Vc = \frac{1}{3}\pi r2h = \frac{1}{3}A{Base}h$$
• Surface Area of a Pyramid: $$Sp=A{Base}+nA_{triangle}$$, where n is the number of sides of the regular polygon that makes up the base
• Surface Area of a Cone: $$S_{c}=\pi r2+\pi rs$$, where r is the radius and s is the slant height
• Surface area of a sphere: $$S_{s}=4\pi r2$$
• Volume of a sphere: $$V_{s}=\frac{4}{3}\pi r3$$ ## Trigonometric Geometry In right-angle trigonometry the mnemonic SOHCAHTOA is true
• $$\sin\theta = \frac{\text{opposite side}}{\text{hypotenuse}}$$

• $$\cos\theta = \frac{\text{adjacent side}}{\text{hypotenuse}}$$

• $$\tan\theta = \frac{\text{opposite side}}{\text{adjacent side}}$$
Also, we know the following:

• Pythagorean Theorem: $$a2+b2=c2$$

• $$360˚ = 2\pi \text{ rad}$$

• $$s=r\theta$$, where $$s$$ is the length of an arc, $$r$$ is the radius, and $$\theta$$ is the angle from the center

• The area of a sector: $$A_{s}=\frac{1}{2}r2\theta$$, where $$\theta$$ is the angle from the center of the circle measured in radians

## Trigonometric Identities

Law of Cosines: $$c2=a2+b2-2ab\cos C$$
Law of Sines: $$\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}$$
Algebraic definition of tangent: $$\tan x =\frac{\sin x}{\cos x}$$

Definition of reciprocal trigonometric functions:

• $$\sec\theta =\frac{1}{\cos\theta}$$
• $$\cosec\theta =\frac{1}{\sin\theta}$$
• $$\cot\theta =\frac{1}{\tan\theta}$$

The main identities

• $$\sin2\theta+\cos2\theta=1$$
• $$1+\tan2\theta=\sec2\theta$$
• $$1+\cot2 \theta = \cosec2 \theta$$

Compound Angle Identities

• $$\sin(A\pm B)=\sin A\cos B \pm\sin B \cos A$$
• $$\cos(A\pm B)=\cos A\cos B \mp\sin A \sin B$$
• $$\tan(A\pm B)=\frac{\tan A \pm \tan B}{1 \pm \tan A \tan B}$$

Double Angle Identities

• $$\sin 2\theta=2\sin\theta\cos\theta$$
• $$\cos 2\theta=\cos2\theta-\sin2\theta=2\cos2\theta-1=1-2\sin2\theta$$
• $$\tan 2\theta = \frac{2\tan\theta}{1-\tan2\theta}$$

Some Trigonometric Transformations

• $$\sin(\pi-\theta)=\sin\theta$$
• $$\cos(\pi-\theta)=-\cos\theta$$
• $$\tan(\pi-\theta)=-\tan\theta$$
• $$\sin(\pi+\theta)=-\sin\theta$$
• $$\cos(\pi+\theta)=\cos\theta$$
• $$\tan(\pi+\theta)=\tan\theta$$
• $$\sin(2\pi-\theta)=-\sin\theta$$
• $$\cos(2\pi-\theta)=\cos\theta$$
• $$\tan(2\pi-\theta)=-\tan\theta$$ Unit Circle
• $$\sin\theta = \text{y-coordinate}$$
• $$\cos\theta = \text{x-coordinate}$$ ## Derivatives of Trigonometric Functions Basic trigonometric derivatives
• $$f(x)=\sin x \rightarrow f'(x)=\cos x$$
• $$f(x)=\cos x \rightarrow f'(x)=-\sin x$$
• $$f(x)=\tan x \rightarrow f'(x)=\sec2 x$$ Derivatives of inverse trigonometric derivatives
• $$f(x)=\arcsin x \rightarrow f'(x) = \frac{1}{\sqrt{1-x2}}$$
• $$f(x)=\arccos x \rightarrow f'(x) = -\frac{1}{\sqrt{1-x2}}$$
• $$f(x)=\arctan x \rightarrow f'(x) = \frac{1}{1+x2}$$ Derivatives of reciprocal trigonometric functions
• $$f(x)=\sec x \rightarrow f'(x)=\sec x \tan x$$
• $$f(x)=\csc x \rightarrow f'(x)=-\csc x \cot x$$
• $$f(x)=\cot x \rightarrow f'(x)=-\csc2 x$$ ## Graphs of Trigonometric Functions Let $$f(x)=a\sin[b(x+c)]+d$$. To calculate the amplitude use: $$a=\frac{\text{maximum value}-\text{minimum value}}{2}$$ To calculate the vertical shift use: $$d=\frac{\text{maximum value}+\text{minimum value}}{2}$$ ## Tangents and Normals For a curve $$f(x)$$, at a specified point $$(x1, y1)$$:
• Equation of the tangent: $$y=f'(x1)(x-x1)+y_1$$
• Equation of the tangent: $$y=\frac{-1}{f'(x1)}(x-x1)+y_1$$