This document provides notes on trigonometric ratios for angles in standard position: 1) It defines positive and negative angles based on clockwise and counterclockwise rotation from the positive x-axis. Co-terminal angles end up in the same standard position. 2) Trig ratios are defined on the unit circle, where the radius is 1. This allows trig ratios to be determined regardless of circle size. 3) Reciprocal trig functions are introduced: cosecant, secant, and cotangent. Examples are given of determining reciprocal functions from given ratios. 4) Three question types are outlined: finding ratios given a point, finding other ratios given one ratio, and determining ratios of

1. 6.1 trignotes.notebook November 01, 2012
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Positive and Negative Angles
If an angle rotates in a COUNTER­clockwise direction from
the positive x axis, it form a POSITVE ANGLE in STANDARD position.
Chapter 6: Trigonometry
6.1 Trig Ratios for angles in standard position.
In Grade 11 you learned the exact values of sine,
cosine, and tangent for 30°, 45°, 60°, and 90° angles
from [0° , 360°]. This year we will expand on Notice the angles can be greater than 360°.
that concept and learn a new way to identify angles.
When the angle rotates clockwise, then we have a negative angle.
Again, it can be more than ­360°.
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2. 6.1 trignotes.notebook November 01, 2012
Angles that end up in the same standard position are called
CO­TERMINAL ANGLES. This means the arm is in the
same position. There are an INFINITE number of positive and
negative co­terminal angles for any angle.
Ex) Give 2 positve and 2 negative angles that are co­terminal to
40°.
Ex) Write a negative angle that is co­terminal to 100°.
Ex) Write a positive angle that is co­terminal to ­19°.
Ex) What reference angle in Quadrant 1 is in the same position
as 500°.
Ex) Write a statement for all angles co­terminal to 30°.
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Trig ratios on the unit cirlce.
Ex) P(3,4) is a terminal point of an angle in standard position.
Find the values of sine, cosine, and tangent.
Ex) P(1,­3) is a terminal point of an angle in standard position.
Find the values of sine, cosine, and tangent.
Remember that:
Therefore, from the above diagram; using x, y, and r
if follows that:
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3. 6.1 trignotes.notebook November 01, 2012
Each of these three trig ratios, has a reciprocal.
The reciprocal of sine is cosecant.
The reciprocal of cosine is secant.
The reciprocal of tangent is cotangent.
Therefore:
Ex) If the value of sin = 0.5, what is the value of csc ?
If the value of cot = ­3, what is the value of tan ?
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QUESTION TYPES:
1) Given a point location, determine the trig ratios
. 2) Given ONE ratio, find other ratios, and angles
Ex)
Ex) If and is in Quadrant 2, find the values of the
other 5 trig functions and the possible values of from [0°,360°]
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3) Given an angle, determine the TRIG RATIOS
Note: a) find the co­terminal angle for 510°. Do you remember the EXACT
values of SINE, COSINE, and TANGENT for this angle?
b) use your calculator to enter the NEGATIVE angles and find the ratios
using it INVERSE feature.
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4. 6.1 trignotes.notebook November 01, 2012
Notice that the size of the circle is NOT important in trig ratios.
.Since the triangle are SIMILAR, the results are the same!
If we make the radius of the circle become "1" (known as
the UNIT CIRCLE), then:
13
Now sin = 1
5
cos = 12
tan =
You MUST remember this!!
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HOMEWORK: Pg. 474 #3,4,5,
6 (from Grade 11), 7a), 8, 9,10.
11a)b), Mult. Choice #1,2.
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