Simplifying trigonometric expressions
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1. Simplifying trigonometric expressions may require trial and error but can be helped by using identities, writing functions in terms of sines and cosines, getting a common denominator, factoring, and breaking fractions. 2. An example problem is simplified by factoring out a common cos x term, using the identity cos^2 x = 1 - sin^2 x, and applying exponent properties to get the final expression of cos^3 x. 3. Strategies presented include using trigonometric identities to write functions in standard form, getting a common denominator, using trigonometric angle addition identities, and breaking fractions.
Simplifying trigonometric expressions
- 2. Simplifying trigonometric expressions often takes some trial and error, but the following strategies may be helpful: Use algebra and fundamental identities to simplify the expression. Sometimes, writing all functions in terms of sines and cosines may help. Sometimes, combining fractions by getting a common denominator may help. Sometimes, breaking one fraction into two fractions may help: a+b/c = a/c + b/c Sometimes, factoring may help
- 3. Example: tan x sec x= = sin x . cos x cos x 1 = sin x sin x cos x 1 cos x • tan x = sin x / cos x • sec x = 1/ cos x • To divide by a fraction, multiply by the reciprocal of the denominator • Reduce the resulting product ANSWER
- 4. Factor out a common factor of cos x Use the identity: cos2 x = 1-sin2 x Use a property of exponents to multiply cos x and cos2 x = cos x (1-sin2 x) = cos x . cos2 x = cos3 x cos x – cos x sin2x ANSWER
- 5. = sin x +cos x . = = sin2 x + cos2 x sin x sin x cos x sin x • cot x = cos x / sin x • Get a common denominator of sin x and add the two fractions • sin2 x + cos2 x =1 • csc x = 1/ sin x sin2 x + cos2 x sin x 1 sin x csc x = = sin x + cos x cot x ANSWER
- 6. sec x – cos x sec x sec x sec x - cos x sec x = = 1- cos2 x = sin2 x • a+b/c = a/c + b/c • sec x divided by itself is 1 • cos x/ sec x = cos2 x • 1- cos2 x = sin2 x ANSWER
- 7. SIMPLIFYING THE FOLLOWING: 1. CSC X –SIN X CSC X 2. SIN X + COS X COS X 1+ SIN X
- 8. Give at least 3 examples of word problems involving trigonometric expressions and solutions. Write it in a whole sheet of paper which will be submitted next meeting. Good bye class!