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Trigonometric Identities and Equations
Unit Objectives: Verify trigonometric identities are true. Solve trigonometric equations Application problems with trigonometric functions and ratios. Todayβs Objective: I can verify trigonometric identities.
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An equation that is true for all values of x or π.
Identity: Example: An equation that is true for all values of x or π. π₯ 5 π₯ 3 = π₯ 2 where π₯β 0 Domain of validity Basic Identities: Tangent Identities: Reciprocal Identities: sin π cos π tan π = cot π = cos π sin π 1 1 1 sec π = cot π = csc π = cos π tan π sin π 1 1 1 cos π = tan π = sin π = sec π cot π csc π
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Domain of validity: Domain of validity:
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State the domain of validity and verify the identity
Pick a side Change to sin or cos Simplify to other side π₯β Β± π 2 ,Β± 3π 2 ,β¦ ( sin π₯) ( sec π₯) = tan π₯ =( sin π₯) 1 cos π₯ = sin π₯ cos π₯ ( sin π₯) ( sec π₯) = tan π₯ ( sin π₯) cot π₯ = cos π₯ π₯β 0,Β±π,Β±2π,β¦ =( sin π₯) cos π₯ sin π₯ ( sin π₯) cot π₯ = cos π₯ 1 sin π₯ csc π₯ sec π₯ = cot π₯ csc π₯ sec π₯ = 1 sin π₯ β
cos π₯ 1 = cos π₯ sin π₯ = cot π₯ = 1 cos π₯ π₯β 0,Β± π 2 ,Β±π,β¦
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Pythagorean Identities
1+ tan 2 π= sec 2 π π₯ 2 + π¦ 2 =1 =1+ sin 2 π cos 2 π 1+ tan 2 π cos 2 π + sin 2 π =1 = cos 2 π cos 2 π + sin 2 π cos 2 π = = cos 2 π + sin 2 π cos 2 π 1 y ΞΈ x = 1 cos 2 π = sec 2 π 1+ cot 2 π= csc 2 π
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Simplifying an Expression
Verifying an identity tan 2 π β sin 2 π = tan 2 π sin 2 π = sin 2 π cos 2 π β sin 2 π = sin 2 π cos 2 π β sin 2 π cos 2 π cos 2 π tan 2 π β sin 2 π = sin 2 π β sin 2 π cos 2 π cos 2 π = sin 2 π (1β cos 2 π ) cos 2 π = sin 2 π sin 2 π cos 2 π = tan 2 π sin 2 π Simplifying an Expression = 1 sin π β sin π cos π = 1 cos π csc π tan π = sec π Pg. 908 #9-60 by 3s
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Trigonometric Identities Day 2
Todayβs Objective: I can verify identities.
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Trigonometric Identities
csc π = 1 sin π ππ¨π¬ π π½ + π¬π’π§ π π½ =π sin 2 π =1β cos 2 π cos 2 π =1β sin 2 π sec π = 1 cos π π¬ππ π π½ =π + πππ§ π π½ tan 2 π = sec 2 π β1 tan π = sin π cos π sec 2 π β tan 2 π =1 ππ¬π π π½ =π+ ππ¨π π π½ cot π = cos π sin π cot 2 π = csc 2 π β1 csc 2 π β cot 2 π =1
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