$Decorative banner$

Geometry & Trigonometry

1 / 15

What is the distance formula in 3D space?

Flip

Spacebar

For points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ , the distance $d$ is given by $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.$

Flip

Spacebar

All flashcards

What is the distance formula in 3D space?

For points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the distance $d$ is given by $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.$$

How do you calculate the midpoint in 3D space?

The midpoint $M$ between points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right).$$

What is the volume formula for a cube?

The volume $V$ of a cube with side length $s$ is given by $$V = s^3.$$

What is the surface area formula for a rectangular prism?

The surface area $SA$ of a rectangular prism with dimensions $l$, $w$, and $h$ is given by $$SA = 2(lw + lh + wh).$$

What is the volume formula for a sphere?

The volume $V$ of a sphere with radius $r$ is given by $$V = \frac{4}{3}\pi r^3.$$

What is the surface area formula for a cylinder?

The surface area $SA$ of a cylinder with radius $r$ and height $h$ is given by $$SA = 2\pi r^2 + 2\pi rh.$$

What defines parallel lines in 3D geometry?

Parallel lines are lines that never intersect and maintain a constant distance between them.

What is the sine rule for triangles?

The sine rule states that $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ for sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively.

What is the cosine rule for triangles?

The cosine rule states that $$c^2 = a^2 + b^2 - 2ab \cos C$$ for sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively.

What is the definition of one radian?

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

What are the key conversions between degrees and radians?

$180° = \pi$ radians and $1 \text{ radian} \approx 57.3°.$

What is the Pythagorean identity in trigonometry?

The Pythagorean identity states that $$\sin^2 \theta + \cos^2 \theta = 1.$$

What is the amplitude and period of the sine function?

For the sine function $y = \sin x$, the amplitude is 1 and the period is $2\pi$.

What is the general form of transformations for trigonometric functions?

The general form is $$y = A \sin(B(x - C)) + D$$ where $A$ affects amplitude, $B$ affects period, $C$ affects horizontal shift, and $D$ affects vertical shift.

What are the steps to solve trigonometric equations?

1. Isolate the trigonometric term. 2. Use identities to simplify. 3. Use inverse functions to solve. 4. Consider all solutions within the domain.

$Decorative banner$

Geometry & Trigonometry

1 / 15

What is the distance formula in 3D space?

Flip

Spacebar

For points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ , the distance $d$ is given by $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.$

Flip

Spacebar

All flashcards

What is the distance formula in 3D space?

For points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, the distance $d$ is given by $$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}.$$

How do you calculate the midpoint in 3D space?

The midpoint $M$ between points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$ is given by $$M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}, \frac{z_1+z_2}{2}\right).$$

What is the volume formula for a cube?

The volume $V$ of a cube with side length $s$ is given by $$V = s^3.$$

What is the surface area formula for a rectangular prism?

The surface area $SA$ of a rectangular prism with dimensions $l$, $w$, and $h$ is given by $$SA = 2(lw + lh + wh).$$

What is the volume formula for a sphere?

The volume $V$ of a sphere with radius $r$ is given by $$V = \frac{4}{3}\pi r^3.$$

What is the surface area formula for a cylinder?

The surface area $SA$ of a cylinder with radius $r$ and height $h$ is given by $$SA = 2\pi r^2 + 2\pi rh.$$

What defines parallel lines in 3D geometry?

Parallel lines are lines that never intersect and maintain a constant distance between them.

What is the sine rule for triangles?

The sine rule states that $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ for sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively.

What is the cosine rule for triangles?

The cosine rule states that $$c^2 = a^2 + b^2 - 2ab \cos C$$ for sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively.

What is the definition of one radian?

One radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle.

What are the key conversions between degrees and radians?

$180° = \pi$ radians and $1 \text{ radian} \approx 57.3°.$

What is the Pythagorean identity in trigonometry?

The Pythagorean identity states that $$\sin^2 \theta + \cos^2 \theta = 1.$$

What is the amplitude and period of the sine function?

For the sine function $y = \sin x$, the amplitude is 1 and the period is $2\pi$.

What is the general form of transformations for trigonometric functions?

The general form is $$y = A \sin(B(x - C)) + D$$ where $A$ affects amplitude, $B$ affects period, $C$ affects horizontal shift, and $D$ affects vertical shift.

What are the steps to solve trigonometric equations?

1. Isolate the trigonometric term. 2. Use identities to simplify. 3. Use inverse functions to solve. 4. Consider all solutions within the domain.