Download Lesson 12.4 - James Rahn

Section 12-4
Then you applied
these to several
oblique triangles
and developed
the law of sines
and the law of
cosines.
Now we’ll extend the definitions of
trigonometric ratios to apply to any size
angle.


In this investigation you’ll learn how to
calculate the sine, cosine, and tangent of
non-acute angles on the coordinate plane.
Procedural Note
1. Draw a point on the positive x-axis. Rotate the
point counterclockwise about the origin by the
given angle measure and draw the image point. (If
the angle measure is negative, rotate clockwise.)
Then connect the image point to the origin. The
angle between the segment and the positive xaxis, in the direction of rotation, represents the
amount of rotation.
2. Use your calculator to find the sine, cosine, and
tangent of this angle.
3. Estimate the coordinates of the rotated point.
4. Use the distance formula to find the length of the
segment.
d
 x2  x1 
2
  y2  y1 
2


Follow the Procedure Note for each angle
measure given below.
An example is shown for 120°.
a. 135°
b. 210°
c. 270°
d. 320°
e. 100°

Experiment with the estimated x- and ycoordinates and the segment length to find a
way to calculate the sine, cosine, and tangent.
(Your values are all estimates, so just try to
get close.)



Plot the point (3, 1) and draw a segment from
it to the origin. Label as A the angle between
the segment and the positive x-axis.
Use your method from Step 2 to find the
values of sin A, cos A, and tan A.
What happens when you try using the inverse
sin-1 to find the value of A? What happens for
cos-1 and tan-1 ?


Now consider the general case of the angle
between the positive x-axis and a segment
connecting the origin to the point (x, y).
Give definitions for the values of sin θ, cos θ ,
and tan θ.



Suppose you know that sin B = 0.47.
That information isn’t enough to determine
whether B is about 28°, 151°, or even 208°.
You need additional information to
determine the measure of angle B.
When you draw the point (3, 1)
and make the angle with the
origin you notice that
sin A 
cos A 
1
10
3
10
1
tan A 
3
But the calculator gives:
One matches with
step 3.
•
•
•
You can use a graph to find the
angle.
Create a right triangle by drawing
a vertical line from the end of the
segment to the x-axis. Use right
triangle trigonometry to find the
measure of the angle with its
vertex at the origin.
The acute angle in this reference
triangle, labeled B, is called the
reference angle.
•
•
•
Use the measure of the reference
angle to find the angle you are
looking for.
In this case, angle B has measure
18.435°.
Angle A measures 180° 18.435°, or 161.565°.
Find sin 300° without a
calculator.
• Rotate a point counterclockwise 300°
from the positive x-axis. The image
point is in Quadrant IV, 60° below the xaxis. The reference angle is 60°.
• The sine of a 60° angle is 3 .
2
• The sine of an angle is the y-value
divided by the distance from the
origin to the point.
• Because the y-value is negative in Quadrant
IV,
3
sin 300° =
2

What measure describes an angle, measured
counterclockwise, from the positive x-axis to
the ray from the origin through (4, 3)?


A reference triangle constructed with a
perpendicular to the x-axis has sides 3-4-5.
The reference angle has measure tan-1 = 3/4
or 36.9°. The graph shows that is more than
180°, so 180° 36.9°, or 216.9°.
The graph shows that is more than 180°, so
180° + 36.9°, or 216.9°.