Download Trigonometry 2 - Trig Ratios

Math 20-1
Trigonometry Lesson #2
Trigonometric Ratios of Angles in Standard Position
Objective: By the end of this lesson, you will be able to:
Recall:
For right triangles:
 Pythagorean Theorem:

Trigonometric Ratios:
tan  
sin  

cos  
This year, we will extend the definition of the trigonometric ratios so that we can find tangent,
sine, and cosine of any angle, not just acute angles.
Notice that for any angle in standard position, if you draw a vertical line from the terminal arm to
the x-axis, you get a _____________ triangle in which one angle is the ____________________
angle.
e.g. 1) Sketch an angle in standard position that has the point P(4, 3) on its terminal arm. Label
the distance from the origin to P with the letter r. Then draw a vertical line from P to the
x-axis.
a) What are the lengths of the sides opposite and adjacent to the angle  ?
b) Determine the length of the hypotenuse, r.
Math 20-1
Trigonometry Lesson #2
c) Determine the three primary trig ratios of the angle  .
d) Determine the measure of the angle  , to the nearest degree.
In general, given a point P(x, y) on the terminal arm of an angle in standard position:
• P(x, y)
r
tan  

sin  
cos  
* This works even when the terminal arm is not in Quadrant I. We just have to keep the sign of
the coordinates in mind.
e.g. 2) The point (-7, 4) lies on the terminal arm of an angle,  , in standard position. Sketch the
angle, and determine the exact trigonometric ratios of the angle  .
* Notice: __________ and ___________ are negative, because the x-coordinate of the
point on the terminal arm is negative.
* The r value will always be ________________, no matter which direction it is pointing.
Math 20-1
Trigonometry Lesson #2
Summary: CAST Rule Foldable
e.g. 3) Determine the sign of all three trig ratios for the angle 97 in standard position.
e.g. 4) If sin   0 and tan   0 , in which quadrant does the terminal arm of  lie?
e.g. 5) a) Suppose  is an angle in standard position with terminal arm in quadrant IV and
2
sin    . Determine the exact values of cos and tan  .
5
b) If you didn’t have the information that the terminal arm of  was in quadrant IV,
which other quadrant could it have been in? What would the values of cos and
tan  be in this case?
* For every trig ratio, there are _________ angles between 0 and 360 with that ratio.
Assignment:
p. 96-99 #3-6, 8, 11, 14, 16, 24, 28
For a challenge: #22, 23