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2312 - Section 4.4 - Trigonometric Expressions and Identities
In this section we are going to practice the algebra involved in working with the trigonometric functions. Notational Conventions
1. An expression such as sin π really means sin(π). An exception to this, however, occurs in expressions such as sin(π΄ + π΅), where the parentheses are necessary. Example: sin (π
4+ π2) β sin π
4+ π2
2. Parentheses are often omitted in multiplication. For example: (sin π)(cos π) is usually written sin π cos π.
3. The quantity (sin π)π = sinπ π = sin π sin π β¦ sin πβ π π‘ππππ
.
Example: (sin π)2 = sin2 π = sin π sin π. sin2 π β sin π2
In simplifying expressions, it may be useful to use the following identities. Basic Trigonometric Identities Reciprocal Identities
1. sec π = 1cosπ
, cos π β 0
csc π = 1sinπ
, sin π β 0
cot π = 1tanπ
, tan π β 0
2. sinπcosπ
= tan π ; cosπsinπ
= cot π Pythagorean Identities
3. sin2 π + cos2 π = 1
tan2 π + 1 = sec2 π
cot2 π + 1 = csc2 π
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Example: Simplify.
(sin πΌ β cos πΌ)2 + 2 sin πΌ cos πΌ
Recall: (π + π)2 = π2 + 2ππ + π2 (π β π)2 = π2 β 2ππ + π2
Example: Simplify.
(1 β cosπΌ)(csc πΌ + cot πΌ) Example: Simplify.
sin4 π β 2 sin2 π + 1
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Example: Simplify. sin πΌ (cot πΌ + tanπΌ)
Example: Simplify.
cos2 πΌ + sin2 πΌ + cot2 πΌ
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Example: Simplify. 3 cos2 πΌ + sec2 πΌ + 3 sin2 πΌ β tan2 πΌ
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Example: Simplify. tanπΌ
sec πΌ + 1+
tanπΌsec πΌ β 1
tanπΌsecπΌ+1
+ tanπΌsecπΌβ1
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Example: Simplify.
1 βcos2 πΌ1 β sin πΌ
