Math 143 Review for Quiz 14 page 1

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1 Math Review for Quiz age. Solve each of the followig iequalities. x + a) < x + x c) x d) x x + <. Solve each of the followig equatios. e) x x + a) x + x x + + x + c) x + x + d) x + 0 x + x. Simlify each of the followig. a) log + log log +log c) log d) l e e) e l f) log log g) log M h) log M i) log M j) log () u) log A log A A + logb B v) l log e 0 (0:00) + log 9 k) log 9 + log l) log 9+log m) log ( ) ) e l A + e l B o) el A+l B ) log (si ) w) log x + log y x) log x+log y q) log sec 0 r) log (ta 0 ) s) log 9 t) log. Simlify each of the followig. a) log log c) log d) log (cos ) e) log si. Simlify each of the followig. a) log 0 + log 0 log log c) log log d) log + log log e) log + log log f) log 0: + log :. Write each of the followig as a sigle logarithm. Assume that A ad B rereset ositive umbers. a) log A + log B log A + log B c) log A log B d) log A + log B e) + log A f) log 0 g) l h) log A i) log A. a) Comute the area of the triagle determied by the oits A (; ), B (0; ), ad C (0; ). Comute the area of the triagle determied by the oits A (log x; log y), B (log x; log y), ad C (log x; log y).. a) Suose that is NOT a acute agle. Fid the exact value of cos ad ta if si. Ratioalize the deomiator i your aswers. Suose that is a agle with 90 < < 0. Fid the value of si ad cos if ta. Ratioalize the deomiator i your aswer. c) Suose that is a agle ot i the third quadrat. Fid the value of si, cos, ad ta if sec. Ratioalize the deomiator i your aswer.

2 Math Review for Quiz age 9. Suose that ta x. Comute si x. 0. Simlify each of the followig. a) si cos c) ta ( ) d) ta e) sec. Simlify each of the followig. (Write it i terms of si, cos ; ad ta. f) csc a) si ( + 0 ) c) ta ( 0 ) e) cos ( ) g) si ( 0 ) cos ( + 0 ) d) si (0 ) f) ta ( ) h) ta (0 ). Cosider the exressio cos (0 ). All of the followig exressios are equal to cos (0 ) ; excet for oe. Which oe? A) cos B) si (90 ) C) si ( 90 ) D) cos E) cos ( ). Simlify each of the followig. a) log (ta 0 ) + log (si ) l (ta ) sec 0 csc + cot 0 ta 0 ta c) + ta 0 ta d) ta 0 ta 0. Prove that if is ay acute agle, the si + cos >.. a) Comute the exact value of ta 0 + ta 0 Comute the exact value of ta 0 c) Based o your digs, determie whether the followig statemet is true or false: ta ( + ) ta + ta. Comute the exact value of each of the followig. Ratioalize deomiators ad simlify your aswer. a) cos si c) ta. Prove each of the followig co-fuctio idetities. For arts a) ad, use a aroriate comud agle formula. Why is that ot a otio for art c)? a) si cos cos si c) ta cot. Suose that ad are acute agles with si the followig. ad cos. Comute the exact value of each of a) cos d) ta g) si ( + ) j) cos ( ) ta e) si h) si ( ) k) ta c) si f) cos i) cos ( + ) l) ta 9. Suose that si A. Comute the exact value of each of the followig. a) cos A ta A c) si A d) cos A 0. Suose that si B ad B is ot i the rst quadrat. Comute the exact value of each of the followig. a) sec B ta B c) cos B d) ta B

3 Math Review for Quiz age. The exressio si x si x ta x is equivalet to which of the followig? A) ta x B) cot x C) ta x D) cot x E) sec x. Comute the exact value of ta ta + ta ta. (Hit: there is a easy way ad also a di cult way to do this.). Simlify each of the followig. Preset exact values. Ratioalize the deomiator. a) si ta + si cos + cos + cos + ::::: + cos 9 + cos 0. Solve each of the followig equatios. You may reset your aswer i degrees. a) si x c) ta x d) si x 0 e) f) ta x g) cos h) si. Solve each of the followig equatios. a) ta x + ta x 0 si x c) si x si x d) si x + cos x e) si x. Solve each of the followig equatios. Preset exact values of all aswers. a) log (x + ) x c) log (x ) d) 0:x 0. Solve each of the followig equatios. Make sure to check your solutios. a) log (x ) + log (x ) log ( x) + log ( x ) c) log (x + ) + log (x ) d) log (x ) log (x + 0) e) log (x + ) log (x + ) f*) log x (x + 0) log x (x ). Circle C has a radius uit log. Circle C has a radius uit log. The ceters are at a distace of uits from each other. We draw the lies taget to both circles. a) Fid a aroximatio of the agle formed by the two taget lies. Comute the distace betwee the two oits of tagecy o oe of the commo taget lies. 9. Cosider the arabola y x x. a) Fid both coordiates of the vertex. Fid all x itercets. Preset exact values. 0. Seattle, WA ad Sa Fracisco, CA are located aroximately o the same logitude. The latitude of these cities are : N ad : N: Fid the distace betwee the two cities assumig that the Earth is a shere with radius 90 miles. Roud your aswer to the earest mile.. The latitude of Seattle, WA is : N. A satellite above Seattle aears statioary i the sky. Fid the seed of the satellite if it is at a height of 00 miles above the surface of Earth. Assume that the Earth is a shere with radius 90 miles.

4 Math Review for Quiz age. Fid the domai for each of the followig fuctios. a) f (x) x + g (x) x + c) h (x) log (x + ) d) f (x) x x e) g (x) x x f) h (x) l x x g) f (x) x x + 9 h) f (x) x x i) f (x) l x 9 + j) f (x) ta x x. For each of the followig airs of grahs, d the coordiates of all oits where they itersect. a) y x + x ad y x (x + ) + (y + ) 0 ad (x ) + (y ) 0 c) (x + ) + (y + ) ad x + (y ). Cosider the yramid ABCDE if its base is a square ABCD with sides m ad side AE BE CE DE 0 m: Comute a aroximate value for the agle that is formed betwee a triagular face ad the base.. Prove each of the followig idetities. a) ta x + + si x sec x sec x + ta x si x c) + si x (si x + ). Comute a aroximate value for each of the agles i a triagle with sides cm, cm, ad 0 cm.. Oe umber a is 0 greater tha twice aother umber b: Fid each of the followig. a) The miimal value of a + b. c) The maximal value of b a The miimal value of ab. We drew a -sided regular olygo ito a circle with radius R. I terms of R ad, exress a) the erimeter of the olygo the area of the olygo. 9. Cosider the icture show. Give that B is the ceter of the circle, rove that.

5 Math Review for Quiz age Aswers. a) ( ; ) [ ( ; ) ; [ (; ) c)..) a) ; ( does t work) c) 0; d). a) c) d) e) l) m) udeed ) A + B o) AB ) u). a) v) w) x + y x) xy c) d) 9 ; d) ( ; ) e) ( ; ) f) g) M h) M i) M j) e). a) c) d) e) f) 0. a) log AB log AB A c) log d) log B B A e 9 9 g) l h) log i) log A A q) r) s) t) 0 e) log (A) f) log 0 0 k). a) 0 uit uit. a) cos c) si cos a) ta ta si c) 0 d) udeed e) f) cos. a) si cos c) ta d) si e) cos f) ta g) si h) ta. D. a) 0 c) d). Claim: if is ay acute agle, the si + cos >. Proof: Let a, b, ad c be sides of a right triagle that cotais as a agle. If is the agle oosite side a, the si a c ad cos b. To rove: c Sice c > 0, we ca multily both sides by c. si + cos > a c + b c > a + b c > c c a + b > c This is true for ay triagle by the triagle iequality.

6 Math Review for Quiz age. a) c) false. a) +. a) si cos cos si si cos c) + si cos cos si cos 0 si cos cos cos + si si 0 cos + si si c) ta cot We ca ot use the di erece formula for taget here, because ta Istead, we eed to use a comoud agle formula searately for sie ad cosie (see arts a ad. ta si cos cos si cot is ude ed.. a) c) d) e) 9 f) 9 9 g) + 9 h) + 9 i) a) j) c) d) k) l) a) c) 9 d). A.. a). a) o solutio + k 0 where k Z c) + k 0 where k Z d) k 0 where k Z. a) e) k 0 where k Z f) 0 + k 0 where k Z g) 0 + k 0 where k Z h) 0 + k 0 ad 0 + k 0 where k Z + k ad k where k Z + k where k Z c) k or + k where k Z d) x + k or x + k or x + k where k Z e) x k where k Z. a) ( + log ) c) d) + log 0. a) 9 ( does ot work) ( does ot work) c) ( does ot work) d) 0 e) o solutio f)

7 Math Review for Quiz age. a) 0 0! 9. a) ; ; 0 ad +! ; miles. 9: mi h. a) ( ; ) [ ( ; ) [ ; ) c) ( ; ) d) ( ; 0) [ (0; ) [ (; ) e) [0; ] f) (0; ) g) R h) [; ) [ (; ) i) ( ; ) [ ( ; ) [ (; ) [ (; ) j) x + k where k Z. a) ( ; ) ad (; ) ( ; ) ad (0; ) c) ( ; 0). : 0. a) ta x + + si x sec x sec x + ta x si x LHS c) + si x (si x + ). : ; : ; ad 9: 9. a) 0 0. a) R si RHS ta x + + si x si x + + si x si x ( + si x) + ( + si x) si x + si x + cos x ( + si x) si x si x + si x ( + si x) + si x si x + si x sec x + ta x RHS si x + ( + si x) sec x LHS ( + si x) cos x RHS (si x + ) si x + cos x + si x + si x LHS c) 00 0 R si R cos 0 0 R si cos 0 + si x 9. Lie segmets AB ad BD are both radii i the circle, ad so they are equal. So ABD triagle is isosceles ad so the agles oosite AB ad BD are also equal to each other. Thus ]ADB. The third agle i triagle ADB is 0. Agles ABD ad DBC are sulemetary beause together they form a straight agle. Thus ]ABD + ]DBC subtract add Last revised: March, 0

Lecture Notes Trigonometric Limits page 1

Lecture Notes Trigonometric Limits page 1 Lecture Notes Trigoometric Limits age Theorem : si! Proof: This theorem ad the et oe are ecessary for di eretiatig si ad cos. Recall a theorem: Let r be the radius of a circle. If is measured i radias,