Chapter 4: Trig Functions
4.2 Graphs of Trigonometric Functions
Algebra Refresher
Graph the function, the coordinates of any intercepts, and any maximum or minimum values.
[latex]\displaystyle f(x) = -6 + \dfrac{2}{3} x[/latex]
[latex]\displaystyle g(x) = 4 - \dfrac{3}{2} x[/latex]
[latex]\displaystyle p(t) = t^2 - 4[/latex]
[latex]\displaystyle q(t) = 9 - t^2[/latex]
[latex]\displaystyle F(x) = 2 - \sqrt{z}[/latex]
[latex]\displaystyle G(z) = \sqrt{4 - z}[/latex]
[latex]\underline{\qquad\qquad\qquad\qquad}[/latex]
Algebra Refresher Answers
[latex]\displaystyle (0,-6), ~ (9,0)[/latex]
[latex](0,4){,}[/latex] [latex]~ \left(\dfrac{8}{3},0\right)[/latex]
[latex]\displaystyle (0,-4), ~ (-2,0), ~ (2,0), ~ {Min:}~-4[/latex]
[latex]\displaystyle (0,9), ~ (-3,0), ~ (3,0), ~ {Max:}~9[/latex]
[latex]\displaystyle (0,2), ~ (4,0), ~ {Max:}~2[/latex]
[latex]\displaystyle (0,2), ~ (4,0), ~ {Min:}~0[/latex]
Learning Objectives
- Find coordinates
- Use bearings to determine position
- Sketch graphs of the sine and cosine functions
- Find the coordinates of points on a sine or cosine graph
- Use function notation
- Find reference angles
- Solve equations graphically
- Graph the tangent function
- Find and use the angle of inclination of a line
Location by Coordinates
One of the most useful applications of the trigonometric ratios allows us to find distances or locations specified by angles. Starting with the definitions of sine and cosine,
[latex]\cos \theta = \dfrac{x}{r} ~~~~ {and} ~~~~ \sin \theta = \dfrac{y}{r}[/latex]
we can solve for [latex]x[/latex] and [latex]y{,}[/latex] the coordinates of points on the terminal side of the angle, and obtain the following results.
Coordinates.
If point [latex]P[/latex] is located at a distance [latex]r[/latex] from the origin in the direction specified by angle [latex]\theta[/latex] in standard position, then the coordinates of [latex]P[/latex] are
[latex]{x = r \cos \theta ~~~~ {and} ~~~~ y = r \sin \theta}[/latex]
Example 4.25.
Solution
The location of point [latex]P[/latex] is shown at right. We see that [latex]r = 6{,}[/latex] and we can use a calculator to evaluate [latex]\cos 292°[/latex] and [latex]\sin 292°{.}[/latex]
[latex]x = r \cos 292° {and} y = r \sin 292°\ = 6(0.3746) = 6(-0.9272)\ = 2.2476 = -5.5632[/latex]
The coordinates of [latex]P[/latex] are approximately [latex](2.25, -5.56){.}[/latex]
Checkpoint 4.26.
- Find the cosine of [latex]215°{.}[/latex] How far west should you walk from the big oak in order to be directly north of the treasure?
- Find the tangent of [latex]215°{.}[/latex] How far south should you walk from your present location before you begin digging?
Solution
- 409.58 yds
- 286.79 yds
Bearings
Navigational directions for ships and planes are sometimes given as bearings, which are angles measured clockwise from north. For example, a bearing of [latex]110°[/latex] is equivalent to an angle of [latex]20°[/latex] in standard position, or to its coterminal angle [latex]340°{,}[/latex] as sho