Name Date Class. cot. tan. cos. 1 cot 2 csc 2

Transcription

1 Fundmentl Trigonometric Identities To prove trigonometric identit, use the fundmentl identities to mke one side of the eqution resemle the other side. Reciprocl nd Rtio Identities csc sec sin cos Negtive-Angle Identities Pthgoren Identities sin() sin sin cos cos() cos tn sec As long s ou know this identit, ou cn divide to derive the other two. tn() cot csc Tr the right side since ou cn use reciprocl identities Prove: sin sec cos sin sec cos Modif the right side. Definition of secnt cos Sutrct. Use common denomintor. cos Pthgoren identit sin cos Rewrite the product. sin Definition of tngent sin cos, so sin cos. Write the missing step or reson to prove ech trigonometric identit.. cot() cot Modif the left side. cos ( ) cot sin ( ) cot sin ( ) cos cot sin Negtive-ngle identit cot cot. seccot csc Modif the left side. csc Simplif. csc csc Holt McDougl Alger

2 Retech Fundmentl Trigonometric Identities (continued) Rewriting trigonometric epression is similr to proving trigonometric identit. Use the fundmentl identities to rewrite the epression. Trigonometric Identities csc sin cos sec Rewrite sin() sin sin cos cos() cos tn sec tn() cot csc in terms of sin. sin cos, so cos sin. ( ) ( ) ( ) ( ) Pthgoren identit Fctor the numertor. Simplif. Think: Let sin. ( )( ) sin The epression is in terms of sin. Write the missing step or reson to rewrite ech epression in terms of cos. 3. ( sec )( sin ) sin cos 4. tn cos Multipl. Definition of tngent Use the Distriutive Propert. Holt McDougl Alger

3 Ellipses An ellipse is the set of points in the plne for which the sum of the distnces from two fied points is the sme numer. The two fied points re clled the foci of the ellipse. This is the stndrd form for the eqution of n ellipse with center (0, 0). The vertices of the ellipse re (, 0), (, 0), (0, ), nd (0, ). If >, then the mjor is is horizontl. If >, then the mjor is is vertictl. To write the eqution of the ellipse with center (0, 0) nd vertices (, 0) nd (0, 5), identif the vlues of nd from the vertices. Sustitute nd 5 into the stndrd form for the eqution: 5 Since 5 >, the mjor is of this ellipse is verticl. 4 5 Write n eqution in stndrd form for ech ellipse with center (0, 0).. Vertices (4, 0) nd (0, 6). Vertices (7, 0) nd (0, 3) 3. Vertices (0,0) nd (0,8) 4, 6,, Grph: 4 5 Holt McDougl Alger

4 Hperols The stndrd form for the eqution of hperol with center t (0, 0) depends on whether the grph opens horizontll or verticll. Horizontl Opening Vertices: (, 0) nd (, 0) lie on the -is. Co-vertices: (0, ) nd (0, ) lie on the -is. Verticl Opening Vertices: (0, ) nd (0, ) lie on the -is. Co-vertices: (, 0) nd (, 0) lie on the -is. The grph opens horizontll. Vertices re (, 0) nd (, 0). Co-vertices re (0, 4) nd (0, 4). So, nd 4. Sustitute into the stndrd form eqution nd simplif. 4 6 The grph opens verticll. Vertices re (0, ) nd (0, ). Co-vertices re (4, 0) nd (4, 0). So, nd 4. Sustitute into the stndrd form eqution nd simplif. 4 6 Write n eqution in stndrd form for ech hperol... Grph opens Grph opens Holt McDougl Alger

5 Hperols (continued) To grph hperol with center t (0, 0): Use the stndrd form to determine the direction the hperol opens. Identif nd from the eqution to find nd plot the vertices nd co-vertices. Drw rectngle tht contins the vertices nd co-vertices. Drw nd etend the digonls of the rectngle. These re the smptotes. Drw the hperol. Grph Horizontl Opening Vertices: (, 0) nd (, 0) Co-vertices: (0, ) nd (0, ) Asmptotes:. 5 6 Since the coefficient of is positive, the hperol opens horizontll. Find nd plot the vertices nd co-vertices. 5, so 5. Vertices: (5, 0) nd (5, 0) 6, so 4. Co-vertices: (0, 4) nd (0, 4) Drw the smptotes nd write their equtions. 4 5 Drw the hperol. Asmptotes: Verticl Opening Vertices: (0,) nd (0,) Co-vertices: (,0) nd (,0) If the coefficient of is positive, the hperol is horizontl nd vertices lie on the -is. If the coefficient of is positive, the hperol is verticl nd vertices lie on the -is. The smptotes help sketch the grph. Grph the hperol Grph opens Vertices: Co-vertices: Asmptotes: Holt McDougl Alger

6 Prols A prol is the set of ll points in the plne tht re the sme distnce from fied point, clled the focus, nd from fied line, clled the directri. To write the eqution of prol with verte (0, 0), sustitute the vlue of p into the eqution for the stndrd form. Use p to represent Use if the is of smmetr is the -is. this distnce. Use if the is of smmetr is the -is. p > 0 if the prol opens to the right or upwrd. p 0 if the prol opens to the left or downwrd. From the grph of the prol: The -is is the is of smmetr. Use to write the eqution. The prol opens to the right, so p is positive. The focus is (, 0). It is units w from the verte (0, 0). The directri,, is lso units w from the verte. So, p. Sustituting p, 4 8 Note tht the focus nd the directri re the sme distnce from the verte of the prol. Complete to write the eqution in stndrd form for the prol.. Ais of smmetr:. Eqution form: 3. Sign of p: 4. Focus: 5. Vlue of p: 6. Eqution: Grph: +8=0 Find the eqution of prol: ) focus(0, 3.5), ) directri =3 Holt McDougl Alger