Download 12.4 Notes: Extending Trigonometry

12.4 – Extending Trigonometry
Name ________________________________
In Lesson 12.1, the definitions given for the sine, cosine, and tangent ratios applied to acute angles in
right triangles. In Lessons 12.2 and 12.3, you solved problems using sines and cosines of angles in
oblique triangles. In this lesson you’ll extend the definitions of trigonometric ratios to apply to any size
angle.
For measuring angles on the coordinate plane
you begin at the right and move
counterclockwise. You might think of beginning in
Quadrant I and moving in order through
Quadrants II, III and IV.
In the following problems, you’ll learn to calculate the sine, cosine and tangent of non-acute angles on
the coordinate plane. (Use a protractor to draw the angles)
Draw a 120o angle
Estimate the coordinates of a point on the
terminal side of the angle.
(____, _____)
Find the distance from the point to the origin
using the distance formula.
Find the ratios:
x  coordinate

d
 ___________
y  coordinate

d
 ___________
y  coordinate

x  coordinate
 ___________
Compare these numbers to the sine, cosine and
tangent of the angle:
sin(120)  ___________
cos(120)  ___________
tan(120)  ___________
d = __________
Draw a 210o angle
Estimate the coordinates of a point on the
terminal side of the angle.
(____, _____)
Find the distance from the point to the origin
using the distance formula.
Find the ratios:
x  coordinate

d
 ___________
y  coordinate

d
 ___________
y  coordinate

x  coordinate
 ___________
Compare these numbers to the sine, cosine and
tangent of the angle:
sin(210)  ___________
cos(210)  ___________
tan(210)  ___________
d = __________
Draw a 320o angle
Estimate the coordinates of a point on the
terminal side of the angle.
(____, _____)
Find the distance from the point to the origin
using the distance formula.
Find the ratios:
x  coordinate

d
 ___________
y  coordinate

d
 ___________
y  coordinate

x  coordinate
 ___________
Compare these numbers to the sine, cosine and
tangent of the angle:
sin(320)  ___________
cos(320)  ___________
tan(320)  ___________
d = __________
Plot the point (-3, 1) and draw a segment from it to the origin. Label A as the angle between the
segment and the positive x-axis. Find the values of sin( A), cos( A), and tan( A) .
sin( A)  ___________
cos( A)  ___________
tan( A)  ___________
Use the inverse to find the angle. What happens?
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 (without regard to signs). 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.
You can generalize this process to find the sine, cosine, and tangent of any angle.
Example A: What measure describes an angle, measure counterclockwise, from the positive x-axis
to the ray from the origin through (-4, -3)?
Example B: Without a calculator, determine whether the value is positive or negative.
a. sin(72) - ____________
Quadrant II –
Sine = ______
Cosine = _________
Tangent = _________
Quadrant I –
Sine = ______
Cosine = _________
Tangent = _________
b. cos(115) - ____________
c. sin(210) - ____________
d. cos(315) - ____________
e. sin(-120) - ____________
f. cos(225) - ____________
g. tan(240) - ____________
h. tan(115) - ____________
Quadrant III –
Sine = ______
Cosine = _________
Tangent = _________
Quadrant IV –
Sine = ______
Cosine = _________
Tangent = _________