page 20
sin (–θ |
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cos (–θ ) = cos θ |
tan (–θ |
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sin θ |
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cos (θ – 90° ) = cos (90° – θ ) = etc. |
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sin (θ 1 ± θ 2 ) = |
sin θ 1 cos θ 2 ± cos θ 1 sin θ 2 |
OR |
sin (2θ |
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2 sin θ cosθ |
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cos (θ 1 ± θ 2 ) = |
cos θ 1 cos θ 2 + sin θ 1 sin θ 2 |
OR |
cos (2θ |
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tan (θ 1 ± θ 2 ) = |
tan θ 1 ± tan θ 2 |
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+ tan θ 1 tan θ |
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cot θ |
− |
1 |
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cot (θ 1 ± θ 2 ) = |
1 cot θ 2 + |
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tan θ |
2 ± tan θ 1 |
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sin |
θ |
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1 – cos θ |
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-ve if in left hand quadrants |
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cos |
θ |
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± |
1 + cos θ |
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2 |
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tan θ-- |
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sin θ |
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1--------------------– cos θ |
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1 + cos θ |
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sin θ |
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(cos θ |
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+ |
(sin θ ) = 1 |
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• These can also be related to complex exponents,
cos θ |
= |
ejθ |
+ e–jθ |
sin θ |
= |
ejθ |
– e |
–jθ |
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2 |
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2.3.2 Hyperbolic Functions
• The basic definitions are given below,
page 21
sinh (x ) = |
e-----------------x – e–x |
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hyperbolic sine of x |
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cosh (x ) = |
e-----------------x + e–x |
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hyperbolic cosine of x |
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tanh (x ) = |
sinh (x ) |
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ex |
– e–x |
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hyperbolic tangent of x |
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------------------cosh (x ) |
e----------------- |
x |
+ e |
–x |
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csch (x ) = |
1 |
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2 |
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hyperbolic cosecant of x |
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-----------------sinh (x ) |
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e----------------- |
x |
– e |
–x |
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sech (x ) = |
1 |
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2 |
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------------------cosh (x ) = |
e----------------- |
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+ e |
–x |
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hyperbolic secant of x |
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coth (x ) = |
cosh (x ) |
= |
ex + e–x |
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hyperbolic cotangent ofx |
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------------------sinh (x ) |
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-----------------e |
x |
– e |
–x |
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• some of the basic relationships are,
sinh (–x ) = – sinh (x ) cosh (–x ) = cosh (x ) tanh (–x ) = – tanh (x ) csch (–x ) = – csch (x ) sech (–x ) = sech(x ) coth (–x ) = – coth (x )
• Some of the more advanced relationships are,
page 22
2 |
2 |
2 |
2 |
2 |
2 |
(cosh x ) – (sinh x ) |
= (sech x ) |
+ (tanh x ) |
= (coth x ) |
– (csch x ) = 1 |
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sinh (x ± y ) = |
sinh (x )cosh (y )± cosh (x )sinh (y ) |
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cosh (x ± y ) = |
cosh (x )cosh (y )± sinh (x )sinh (y ) |
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tanh (x ± y ) = |
tanh (x )± tanh (y ) |
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1----------------------------------------------± tanh (x )tanh (y ) |
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• Some of the relationships between the hyperbolic, and normal trigonometry functions are,
sin (jx ) = |
j sinh (x ) |
j sin (x ) = |
sinh (jx ) |
cos (jx ) = cosh (x ) |
cos (x ) = cosh (jx ) |
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tan (jx ) = |
j tanh (x ) |
j tan (x ) = |
tanh (jx ) |
2.3.2.1 - Practice Problems
3. Find all of the missing side lengths and corner angles on the two triangles below,