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IB DP Maths: AA HL复习笔记3.4.2 Exact Values
Category:
IB课程
,
教材笔记
,
福利干货
Date: 2022年7月12日 下午3:10
Trigonometry Exact Values
What are exact values in trigonometry?
For certain angles the values of sin θ, cos θ and tan θ can be written
exactly
This means using fractions and surds
You should be familiar with these values and be able to derive the values using geometry
You are expected to know the exact values of sin, cos and tan for angles of 0°, 30°, 45°, 60°, 90°, 180° and their multiples
The exact values you are expected to know are here:
How do I find the exact values of other angles?
The exact values for sin and cos can be seen on the
unit circle
as the
y
and
x
coordinates respectively
If using the coordinates on the unit circle to memorise the exact values, remember that cos comes
before
sin
The
unit circle
can also be used to find exact values of other angles using symmetry
If you know the exact value for an angle in the first quadrant you can draw the same angle from the x-axis in any other quadrant to find other angles
Remember that the angles are
measured anticlockwise
from the positive x-axis
For example if you know that the exact value for is 0.5
draw the angle 30° from the horizontal in the three other quadrants
measuring from the positive x-axis you have the angles of 150°, 210° and 330°
sin is positive in the second quadrant so sin150° = 0.5
sin is negative in the third quadrant so sin210° = - 0.5
sin is negative in the fourth quadrant so sin330° = - 0.5
It is also possible to find the
negative
angles by measuring
clockwise
from the positive x-axis
draw the angle 30° from the horizontal in the three other quadrants
measuring
clockwise
from the positive x-axis you have the angles of -30°, -150°, -210° and -330°
sin is negative in the fourth quadrant so sin(-30°) = - 0.5
sin is negative in the third quadrant so sin(-150°) = - 0.5
sin is positive in the second quadrant so sin(-210°) = 0.5
sin is positive in the fourth quadrant so sin(-330°) = 0.5
How are exact values in trigonometry derived?
There are two special
right-triangles
that can be used to derive all of the exact values you need to know
Consider a
right-triangle
with a hypotenuse of 2 units and a shorter side length of 1 unit
Exam Tip
Worked Example
转载自savemyexams
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