Additional Identities Trigonometry MATH 103 S. Rook.
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Additional Identities Trigonometry MATH 103 S. Rook
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Transcript of Additional Identities Trigonometry MATH 103 S. Rook.
Additional Identities
TrigonometryMATH 103
S. Rook
Overview
• Section 5.5 in the textbook:– Identities and formulas involving inverse functions– Product to sum & sum to product formulas
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Identities and Formulas Involving Inverse Trigonometric Functions
Identities and Formulas Involving Inverse Trigonometric Functions
• We have discussed how to solve problems involving the inverse trigonometric functions:– e.g. cos(arcsin ½)– Draw and label a right triangle to solve
• Possible for inverse trigonometric functions to appear as arguments in the sum & difference formulas, double-angle formulas, or half-angle formulas– e.g. tan(arccos -½ – arcsin ½)
• Let A = arccos -½ and B = arcsin ½ • Becomes tan(A – B) → difference formula for the tangent
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Identities and Formulas Involving Inverse Trigonometric Functions
(Example)Ex 1: Evaluate without a calculator:
a)
b)
5
2
1sin
2
1tansin 11
3
1sin2cos 1
Product to Sum & Sum to Product Formulas
Product-to-Sum & Sum-to-Product Formulas
• The preceding formulas can be used when we have one angle
• However, situations arise where we wish to operate on two DIFFERENT angles– e.g. Products such as sin A cos B transform to sums– e.g. Sums such as sin A + cos B transform to products
• When considering sines & cosines and two different angles, we have four different situations that can arise
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Product-to-Sum Formulas
• Product-to-Sum Formulas:
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vuvuvu
vuvuvu
vuvuvu
vuvuvu
sinsin2
1sincos
sinsin2
1cossin
coscos2
1coscos
coscos2
1sinsin
Sum-to-Product Formulas
• Sum-to-Product Formulas:
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2sin
2sin2coscos
2cos
2cos2coscos
2sin
2cos2sinsin
2cos
2sin2sinsin
vuvuvu
vuvuvu
vuvuvu
vuvuvu
Product to Sum Formulas (Example)
Ex 2: Express as a sum or difference:
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xx 8sin2cos
Sum to Product Formulas (Example)
Ex 3: Express as a product:
11
xx 3cos5cos
Summary
• After studying these slides, you should be able to:– Apply the different types of formulas learned in
Chapter 5 to the inverse trigonometric functions– Apply the product to sum formulas– Apply the sum to product formulas
• Additional Practice– See the list of suggested problems for 5.5
• Next lesson– Solving Trigonometric Equations (Section 6.1)
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