Trigonometry Worksheet Page 16
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cot =
=
=
.
Rationalize the denominator
H
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Had the angle
been in a different quadrant, negative signs would have been assigned according to the
“ASTC” mnemonic. For example, if
was in quadrant 2 in the above example, the cosine and tangent
results would have been negative, and their reciprocal (the secant and cotangent) results would have
been negative as well. The cosecant result would have remained positive.
For most practical measurement purposes, the cosine, sine and tangent functions are almost always
sufficient. However, in calculus, the reciprocal functions (in particular, the secant function) have many
uses.
Common Identities
There are many useful identities that we use in trigonometry and in calculus. The most common
involving the cosine and sine functions are stated below.
• The Sum and Difference Identities
sin{ ± { = sin cos ± cos sin
cos{ ± { = cos cos ∓ sin sin
Note the reversal of the signs in the cosine sum-difference formula.
Example: Use an appropriate sum-difference formula to determine the exact value of sin{75°{.
Solution: We see that 75° = 45° + 30°. Therefore,
sin{75°{ = sin { 45° + 30° {
= sin { 45° { cos{30°{ + cos { 45° { sin{30°{
$
%
$
#
=
+
st
Referring to the table of 1
Quadrant values
$
$
$
$
$
=
&
These sum-difference identities can be used to create the double-angle identities:
• The Double Angle Identities
sin{2 { = 2 sin cos
cos{2 { = cos
− sin
$
$
The proofs of both are left as exercises XX and YY in the homework.
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Example: If cos =
is in Quadrant 1, determine exact values for sin{2 { and cos{2 {.
and
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Solution: We’ll need the exact value for sin , which we can get from the Pythagorean Identity—it is
$ $
sin =
(You should verify this).
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