Unit Circle and Trigonometric Functions

The video tutorial explains the definitions of sine and cosine starting with right triangles, where sine is the ratio of the opposite side to the hypotenuse, and cosine is the ratio of the adjacent side to the hypotenuse. It highlights the limitation of these definitions to acute angles and extends them using the unit circle, allowing sine and cosine to be defined for any angle. The unit circle, with a radius of one, is introduced in standard position on a coordinate plane, and the intersection points of angles with the circle are used to define sine and cosine values. This approach is applicable to all angles, making it a powerful tool in trigonometry.

What is the definition of cosine in a right triangle?

• Hypotenuse divided by opposite side

• Hypotenuse divided by adjacent side

• Adjacent side divided by hypotenuse

• Opposite side divided by hypotenuse

In a right triangle, what is the sine of an angle defined as?

• Hypotenuse divided by adjacent side

• Opposite side divided by hypotenuse

• Adjacent side divided by hypotenuse

• Hypotenuse divided by opposite side

Why do the right triangle definitions of sine and cosine only work for acute angles?

• Because they require angles to be between 0 and 90 degrees

• Because they require angles to be between 270 and 360 degrees

• Because they require angles to be between 90 and 180 degrees

• Because they require angles to be between 180 and 270 degrees

What is the purpose of extending the definition of sine and cosine in precalculus?

• To include only right angles

• To include only obtuse angles

• To include all angles

• To include only acute angles

What is the initial side of an angle in standard position?

• The side opposite the angle

• The side that remains fixed

• The side that rotates

• The side adjacent to the angle

What is the radius of the unit circle?

• 3

• 2

• 1

• 0

Which point does the unit circle pass through?

• (2, 0)

• (0, 1)

• (0, 2)

• (1, 0)

How is cosine defined using the unit circle?

• As the radius of the unit circle

• As the x-coordinate of the point where the terminal side intersects the unit circle

• As the y-coordinate of the point where the terminal side intersects the unit circle

• As the hypotenuse of the triangle formed

How is sine defined using the unit circle?

• As the radius of the unit circle

• As the x-coordinate of the point where the terminal side intersects the unit circle

• As the hypotenuse of the triangle formed

• As the y-coordinate of the point where the terminal side intersects the unit circle

What is the significance of the unit circle in trigonometry?

• It only works for acute angles

• It only works for obtuse angles

• It only works for right angles

• It allows for the definition of sine and cosine for all angles