Trigonometric Functions¶

Goals for Section 4.1 of the book¶

Students should be able to do each of the following:

Determine an angle's measurement in radians given either the arc length or radius of an arc.
Place an angle in standard position given a verbal description.
Determine the terminal or the initial side of a sector.
Recognize whether an angle is positive, negative or zero.
Recognize when two angles have coterminal sides.
Determine if an angle is acute, obtuse, or a right angle.
Determine if an angle is complimentary or a supplementary angle.
Given any two values of a sectors radius, angle, or arc length determine the value of the remaining quantity.
Determine the area of a sector given the radius and angle.
Given any two values of a sectors radius, angle, or arc length determine the value of the area of the sector.
Recognize when a given angle represents more than one complete
rotation around a circle.

Students should be able to do each of the following:

Determine the coordinate of a point on the unit circle given either the x or the y value for the coordinate.
Given the radian measurement of an angle determine the approximate location of the corresponding point on the unit circle and determine which quadrant the point is in.
Determine the point on the unit circle when the corresponding angle is a multiple of \(\frac{\pi}{4}\).
Use the definition of trigonometric functions to determine the values of the functions given a coordinate on the unit circle.
State the domain and range of the trigonometric functions.
State the fundamental trigonometric identities.
Determine if a basic function is periodic and explain their conclusion in terms of the definition of a periodic function.
Determine the period of a trigonometric function.
Use a calculator to approximate trigonometric functions.

Students should be able to do each of the following:

Determine values of trigonometric functions given right triangles.
Use the Pythagorean identity to determine the value of one trigonometric function in terms of another.
Determine the values of trigonometric functions using multiple right triangles.
Translate a written description of a problem into a trigonometric formulation

Students should be able to do each of the following:

Determine subsets of the domain where sine/cosine are increasing and where they are decreasing.
Determine whether a trigonometric function will increase or decrease given a specific angle.
Determine the reference angle of a given angle in any of the four quadrants.

Students should be able to do each of the following:

Determine the range and domain of the sine and cosine functions.
Determine the amplitude, period, and phase of a trigonometric function given a written or graphical representation.
Determine the vertical shift and scale of a trigonometric function given a written or graphical representation.
Determine the horizontal shift and scale of a trigonometric function given a written or graphical representation.
Graph a sine or cosine wave given a formula or written description.
Determine the formula for a sine or cosine wave given a written description
Determine the formula for a sine or cosine wave given the graph or a written description of the function.
Determine the period of a sine or cosine wave given the graph or a written description of the function.

Students should be able to do each of the following:

Determine the restriction of the domain of a function in order to define an inverse.
State the canonical domain restrictions for sine, cosine, and tangent functions.
Use the definition of the arcsine, arccosine, and the arctangent functions to solve for one value in a trigonometric expression.
Sketch the graphs of the arcsine, arccosine, and the arctangent functions.
Compose trigonometric and inverse trigonometric functions to transform a trigonometric expression into an equivalent form that allows for direct algebraic manipulation to isolate a value in the domain of a trigonometric function in the original expression.
Determine equivalent expressions for the compositions of trigonometric and inverse trigonometric functions so that the results do not include any trigonometric functions.

Students should be able to do each of the following:

State fundamental identities.
State the Pythagorean identitites.
Verify that a given expression is true.
- Using quotient identities
- Bringing together expressions using a common denominator.
- Factoring expressions.
- Using basic properties of trigonometric functions. (E.g.: \(\sin(-\theta)=-\sin(\theta)\).)
Solve for values within expressions that involve logarithms.
Transform algebraic expressions by substituting trigonometric functions.

Students should be able to do each of the following:

Use the sum and difference formulas for the sine function to solve for values within a trigonometric expression.
Use the sum and difference formulas for the cosine function to solve for values within a trigonometric expression.
Verify identities that make use of sum and difference formulas.
Combine the sum and difference formulas with inverse trigonometric functions.