Math Analysis Chapter 5 Notes: Analytic Trigonometric Day 9: Section 5.1-Verifying Trigonometric Identities Fundamental Trig Identities Reciprocal Identities: 1 1 1 sin u = cos u = tan u = cscu secu cot u 1 1 1 csc u = sec u = cot u = sin u cosu tan u Quotient Identities: sin u cosu tan u = cot u = cosu sin u Pythagorean Identities: sin u+ cos u = 1 1+ tan u = sec u 1+ cot u = csc u Even-Odd Identities: sin( u) = sin u cos( u) = cosu tan( u) = tan u csc( u) = cscu sec( u) = secu cot( u) = cot u Practice: Use the given value to evaluate the remaining trig functions. 1. sin x=, tan x< 0 3 Math Analysis Notes Prepared by Mr. Hopp 1
Practice: In -5, Write each expression as a single fraction. Reduce, if possible. sinθ cosθ 1 1. + 3. 1+ cosθ sinθ csc x 1 csc x + 1 4. cot x csc x 5. csc x sin x 1+ cos x + 1+ cosx sinx Practice: In 6-11, Verify the identity algebraically. cos x sin x 6. + = sec x 1+ sinx cosx 7. sin x cos x tan x = Math Analysis Notes Prepared by Mr. Hopp
8. sinθ + cosθ cotθ = cscθ 9. cos x sin x= cos x 1 10. cos x sin x + = 1 11. sec x csc x secθ cosθ = sin θ secθ Day 9: p578#3-45 by 3 s p565#68, 73, 74, 76, 8 Math Analysis Notes Prepared by Mr. Hopp 3
Day 10: Section 5-: Sum and Difference Formulas: After today s lesson you should be able to find the exact value of and angle using the sum and difference formulas. The Sum Formulas: Given two angles θ and β: sin( θ + β) = sinθcos β + cosθsin β cos( θ + β) = cosθ cos β sinθ sin β tan( θ β) sin( θ + β ) + = cos( θ + β ) or tanθ + tan β tan( θ + β) = 1 tanθ tanβ The Difference Formulas: Given two angles θ and β: sin( θ β) = sinθ cos β cosθ sin β cos( θ β) = cosθ cos β + sinθ sin β tan( θ β) sin( θ β ) = cos( θ β ) or tanθ tan β tan( θ β) = 1 + tanθ tanβ Practice: Use the correct formula to evaluate the given trigonometric expression. 1. sin (45 0 30 0 π 5π ). cos + 3 6 13π 3. Find all three trig functions of the given angle: α = using the sum or difference formula. 1 4. Write the equivalent expression as sine, cosine, or tangent of an angle. Then find the exact value of the expression. 5π π 5π π sin cos cos sin 1 4 1 4 Math Analysis Notes Prepared by Mr. Hopp 4
5. Verify the trig identity: sinx = sinxcosx 6. Verify the trig identity: sin( x y) = tan x tan y cos xcos y 7. Find the exact value of the of the following under the given condition: (a) cos(x + y) (b) sin(x + y) (c) tan (x + y) Given: 3 5 sin x =, and x lies in Quad II, and cos y =, and y lies in Quad IV. 5 13 Asn 10: p587#, 4, 14-3(even), 38, 4, 44, 5, 58-64(even) P566#86, 88, 90, 91 Day 11 Quiz on Sections 5-1 and 5- and Graphs of Secant, Cosecant and Cotangent Asn 11 p566#94-110(even) Math Analysis Notes Prepared by Mr. Hopp 5
Day 1 Section 5-3 Half Angle Formulas and Double Angle Formulas: Half Angle Formulas: u 1 u cos =± ( 1+ cosu ) sin =± 1 ( 1 cosu ) u sin u tan = 1+ cos u u 1 cosu tan = sin u u u The signs of sin and cos depend on the quadrant in which u lies. u u u Practice: Calculate sin,cos,and tan for the angle described. 5 3π 1. cos u = ; < u< π 13. 3 3π sin u = ; π < u< 4 Math Analysis Notes Prepared by Mr. Hopp 6
3. 5 π cos u = ; < u< π 13 Practice: Use the half-angle formulas to determine the exact values of the sine, cosine, and tangent of the angle. 1. 195. 9 π 8 Math Analysis Notes Prepared by Mr. Hopp 7
3. 19 π 1 4. 105 Math Analysis Notes Prepared by Mr. Hopp 8
Double Angle Formulas: sin u = sin ucosu cos cos sin u = u u tanu tan u = 1 tan u Practice: Find the exact value of the expression 0 tan105 π π 1.. cos sin 3. 0 1 tan 105 1 1 7π 7π sin cos 1 1 Practice: If sinθ = 4 5 and θ lies in quadrant II, find the exact value of each of the following 4. sinθ 5. cosθ 6. tanθ Asn 1 p597#-(even), 40-58(even) p565#7, 80, 84, 9 Math Analysis Notes Prepared by Mr. Hopp 9
Day 13 Section 5-5 Trigonometric Equations Tips for Solving Trig Equations 1. Get an equation (or equations) in which one trig function equals a constant. Substitute any obvious trig identities 3. Reduce the number of different trig functions (get the equation so it only has 1 trig function). 4. Do any obvious algebra (get terms into 1 fraction, factor, use quadratic formula). Remember, in order to factor the equation MUST be equal to 0!!! 5. Do not divide both members by a variable (you can NOT divide by a trig function). Always find general solution first!!! This is especially important if the angle (x) has a coefficient!!! Examples: Solve the equation (a) giving the general solutions and (b) in the interval [0, π). 1. cos x + 1 = 0. 3 sec x = 0 3. 3csc x 4 = 0 4. sin x = 1 x 5. cos3x 1 = 0 6. cos = Asn 13 p619#-4(even), 6, 30, 34, 38 Math Analysis Notes Prepared by Mr. Hopp 10
Day 14 Section 5-5 Trigonometric Equations with Factoring Examples: Solve the equation (a) giving the general solutions and (b) in the interval [0, π). 1. ( x )( x ) 3tan 1 tan 3 = 0. tan x+ tan x = 0 3. = 4. cosxsinx cosx = 0 sin x sin x 1 0 π π 5. sin x = sin x 6. sin x+ + sin x = 1 4 4 Asn 14 p619#40-7 by 4 s Math Analysis Notes Prepared by Mr. Hopp 11