Solution
Look at the figure at right. An hour and a half represents 1.5 complete rotations, or
[latex]1.5(360°) = 540°[/latex]
Forty minutes is two-thirds of an hour, so the minute hand rotates through
[latex]\dfrac{2}{3}(360°) = 240°[/latex]
Chapter 4: Trig Functions
Evaluate the function.
Look at the figure at right. An hour and a half represents 1.5 complete rotations, or
[latex]1.5(360°) = 540°[/latex]
Forty minutes is two-thirds of an hour, so the minute hand rotates through
[latex]\dfrac{2}{3}(360°) = 240°[/latex]
The volume control on an amplifier is a dial with ten settings, as shown at right. Through how many degrees would you rotate the dial to increase the volume level from 0 to 7?
[latex]252°[/latex]
The degree measure of an angle depends only on the fraction of a whole rotation between its sides, and not on the location or position of the angle. To compare and analyze angles, we place them in standard position so that the vertex of the angle is located at the origin and its initial side lies on the positive [latex]x[/latex]-axis. The figure below shows several angles placed in standard position.
One-half of a complete revolution is [latex]180°{,}[/latex] and three-quarters of one revolution is [latex]270°{.}[/latex] Thus, for angles between [latex]180°[/latex] and [latex]270°[/latex] in standard position, the terminal side lies in the third quadrant, and for angles between [latex]270°[/latex] and [latex]360°{,}[/latex] the terminal side lies in the fourth quadrant.
a. The angle [latex]\alpha[/latex] is one-fifth of a complete revolution, or
[latex]\dfrac{1}{5}(360°) = 72°[/latex]
In standard position, it is a first-quadrant angle, as shown in figure (a) below.
b. The angle [latex]\beta[/latex] is [latex]\dfrac{11}{12}[/latex] of a complete revolution, or
[latex]\dfrac{11}{12}(360°) = 330°[/latex]
In standard position, it is a fourth-quadrant angle. (See Figure [b].)
a. [latex]120°[/latex]
b. [latex]70°[/latex]
In Chapter 3 we defined the sine, cosine, and tangent for obtuse angles by placing the angle in a Cartesian coordinate system. We can do the same for angles that represent rotations.
If [latex]\theta[/latex] is an angle in standard position, and [latex](x,y)[/latex] is a point on its terminal side, with [latex]r = \sqrt{x^2 + y^2}{,}[/latex] then
[latex]{\sin \theta = \dfrac{y}{r}~~~~~~~~~ \cos \theta = \dfrac{x}{r}~~~~~~~~~\tan \theta = \dfrac{y}{x}}[/latex]
We can choose any point on the terminal side of the angle, and the trig ratios defined by its coordinates will be the same. (Can you explain why?)
Because it is the distance from the origin to [latex]P{,}[/latex] [latex]r[/latex] is always positive. However, [latex]x[/latex] and [latex]y[/latex] can be positive or negative (or zero), depending on the angle [latex]\theta{.}[/latex] For example, in the second quadrant, [latex]x[/latex] is negative but [latex]y[/latex] is positive, so the cosine and the tangent of angles between [latex]90°[/latex] and [latex]180°[/latex] are