chapter 5

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5.1

3 pythagorean identities

1) sin2 u + cos2 u = 1 2) 1 + tan2 u = sec2 u 3) 1 + cot2 u = csc2 u

4 fundamental trig. identities

1) Evaluate trigonometric functions 2) Simplify trigonometric expressions 3) Develop additional trigonometric identities 4) Solve trigonometric equations

2 quotient identities

1) tan u = (sin u)/(cos u) 2) cot u = (cos u)/(sin u)

6 cofunction identities

1) sin(/2  u) = cos u 2) cos(/2u)=sinu 3) tan(/2u)=cotu 4) cot(/2  u) = tan u 5) sec(/2u)=cscu 6) csc(/2u)=secu

6 even/odd identities

1) sin(u)=  sinu 2) cos( u) = cos u 3) tan(u)=tan u 4) csc(u)=cscu 5) sec( u) = sec u 6) cot(u)=cotu

6 reciprocal identities

1) sin u = 1/(csc u) 2) cos u = 1/(sec u) 3) tanu=1/(cotu) 4) cscu=1/(sinu) 5) secu=1/(cosu) 6) cotu=1/(tanu)

5.2

an identity is

an equation that is true for all real values in the domain of the variable .

guidelines to verify the trig. identities

1) Work with one side of the equation at a time. It is often better to work with the more complicated side first. 2) Look for opportunities to factor an expression, add fractions, square a binomial, or create a monomial denominator. 3) Look for opportunities to use the fundamental identities. Note which functions are in the final expression you want. Sines and cosines pair up well, as do secants and tangents, and cosecants and cotangents. 4) When the preceding guidelines do not help, try converting all terms to sines and cosines. 5) Always try something! Even making an attempt that leads to a dead end provides insight.

the key to verifying identities and solving equations

the ability to use the fundamental identities and the rules of algebra to rewrite trigonometric expressions .

5.5

product-to-sum formulas

The sum-to-product formulas can be used to rewrite a sum or difference of trigonometric functions as a product . The sum-to-product formulas are: sin u + sin v = sin u  sin v = cos u + cos v = cos u  cos v = 2 sin((u + v)/2) cos((u 2 cos((u + v)/2) sin((u 2 cos((u + v)/2) cos((u 2 sin((u + v)/2) sin((u v)/2) v)/2) v)/2) y)/2)

The product-to-sum formulas are used in calculus to tan u 2 (1   cos u)/(sin u) The signs of sin (u/2) and cos (u/2) depend on = (sin u)/(1 + cos u) the . What you should learn How to use half-angle formulas to rewrite and evaluate trigonometric functions 

power-reducing formulas

The power-reducing formulas are: sin2 u = cos2 u = tan2 u = (1   cos 2u)/2 (1 + cos 2u)/2 (1   cos 2u)/(1 + cos 2u)

The double-angle formulas can be used to obtain the power-reducing formulas

half-angle formulas

sin u       (1  cos u)/2 2 cos u     (1 + cos u)/2 2 quadrant in which u/2 formulas are used in calculus to tan u 2 (1   cos u)/(sin u) The signs of sin (u/2) and cos (u/2) depend on = (sin u)/(1 + cos u)

multiple-angle formulas

The most commonly used multiple-angle formulas are double angle formulas which are listed below: 2u = cos2u= = = 2 sin u cos u cos2 u   sin2 u 2 cos2 u 1 1 2sin2u (2 tan u)/(1   tan2 u) tan 2u = To obtain other multiple-angle formulas, use 4   and 2  or in place of 2   and  in the double-angle formulas or 6 and 3 using the double-angle formulas together with the appropriate trigonometric sum formulas

5.4

reduction formula

a formula involving expressions such as sin( theta +n r/2) or cos (theta + nr/2) where n is an integer that can be derived from sum and difference formulas

difference formulas for sine, cosine and tangent

sin(u + v) = sin u cos v + cos u sin v sin(u  v) = sin u cos v cos u sin v cos(u + v) = cos u cos v sin u sin v cos(u v)=cosucosv+sinusinv tan(u+v)=(tanu+tanv)/(1 tanutanv) tan(u  v) = (tan u  tan v)/(1 + tan u tan v)

5.3

to solve a trig equation of quadratic type

factor the quadratic, or if factoring is not possible, use the Quadratic Formula

preliminary goal in solving trig equations

to isolate the trigonometric function involved in the equation

squaring each side of a trigonometric

this procedure can introduce extraneous solutions, so any solutions must be checked in the original equation to see whether they are valid or extraneous

solutions to sec x = 2

The equation has an infinite number of solutions because the secant function has a period of 2 . Any angles coterminal with the equation’s solutions on [0, 2 ) will also be solutions of the equation.

solving a trig. equation

use standard algebraic techniques such as collecting like terms and factoring