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| Trigonometry |
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Trigonometric functions expressed by the
tangent of the half angle |
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Half angle formulas |
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Trigonometric functions expressed by the
cosine of the double angle |
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Trigonometric identities,
examples |
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| Trigonometric functions expressed by the
tangent of the half angle |
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Replacing a
by a/2
in the above identities, we get
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| Half angle formulas |
| Using
the identities in which trigonometric functions are expressed by
the half angle, |
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applying the definitions of the functions, tangent and cotangent |
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Trigonometric functions expressed by the
cosine of the double angle
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Replacing a/2
by a
in the above identities, we get
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| Trigonometric identities,
examples |
| Example:
Express sin
3x in terms of sin
x. |
| Solution:
Using the sum formula and the double angle formula for the sine function, |
| sin
3x
= sin (2x + x)
= sin 2x · cos x + cos 2x · sin x
= 2sin x cos x · cos x + (cos2 x
-
sin2 x) · sin x |
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= 2sin x · (1 -
sin2 x) + (1 -
2sin2 x) · sin x = 3sin x -
4sin3 x. |
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| Example:
Express tan
3x in terms of tan
x. |
| Solution:
Using the sum formula and the double angle formula for the
tangent function, |
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| Example:
Prove the identity |
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| Example:
Prove the identity |
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Example:
If tan
a
= 3/4, find tan
a/2.
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| Solution:
Use formula |
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to express tan
a/2
in terms of tan
a. |
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| Example:
Prove the identity |
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| Solution:
Substitute |
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then |
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| Example:
Prove the identity |
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| Solution:
Using the formula for the sum of the tangent |
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| Example:
Prove that |
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Solution: Replace
sin a
by cos (p/2
-
a)
and cos a
by sin (p/2
-
a)
and use the sum to product
formula
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| Pre-calculus contents
F |
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| Copyright
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