Precalculus Solutions Review for Test 6 LMCA Section

Transcription

1 Precalculus Solutions Review for Test 6 LMCA Section Memorize all of the formulas and identities. Here are some of the formulas for chapter 5. BasicTrig Functions opp y hyp r sin csc hyp r opp y adj x hyp r cos sec hyp r adj x opp y adj x tan cot adj x opp y Reciprocal Identities sin csc csc sin cos sec sec cos tan cot cot tan Quotient Identities sin tan cos cos cot sin Pythagorean Identities sin cos tan sec cot csc Section 4.5 Know how to use your calculator! Are you in Degrees or Radians?? Graphs of Sine and Cosine State each period and amplitude. Sine and Cosine Graphs y asin( x) y acos( x) Amplitude a Amplitude a Period Period Domain: x Domain: x Range: a y a Range: a y a Note: Please recognize these values.57,.4, 4.7, 6.8 Which mode is your calculator in?? Does my answer mae sense?? Page of 9

2 State each period and amplitude.. y 4sin(5 x) Amplitude = 4 4. Period since 5. 5 This means one full cycle is /5 of a unit long. Note on graphing: To graph sine, we now that y sin xcrosses the x-axis at x = 0, and. y 4sin(5 x) will cross the x-aixs at 5x 0, 5 x, and5x. Solving for x, we get x 0, x, and x y 6cos upside down Period 6 This means one full cycle is 6 units long. Note: Amplitude = 6 6. The (negative) sign reflects the graph accros the x-axis. i.e. Note on graphing: To graph cosine, we now that y cos has a maximum at = 0 and while its minimum is at =. Reflect Graph!!! Since -6 < 0 y 6cos will achieve its minimum value at 0 and ; maximum at. Solving for x, we get min at 0 and. Max at. x. y 5 sin Amplitude = (use decimal for graph). Period 4 This means one full cycle is 4 units long. Note: x x, 4. y cos x Amplitude = Period This means one full cycle is 4 units long. Note: Page of 9

3 Graph at least one full period of each function. (State the amplitude and period) I graphed two periods by mistae. 5. y sin(4 x) 6. y4cos x Amp, Per 4 a = < 0 The graph is refected across the x-axis. Amp 4 4, Per 6 a = < 0 The graph is refected across the x-axis. Section 4.6 Graphs of Tangent, Cotangent, Secant and Cosecant y tan( x) y cot( x) Per ( see graph) 5 Aysmptotes : x,,,,... Per ( see graph) Aysmptotes : x,0,,,... Page of 9

4 Tangent and Cotangent Graphs sin( x) cos( x) y tan( x) y cot( x) cos( x) sin( x) Period Period (n ) Domain:,except ; n n Domain:,except ; n cos( x) 0 sin( x) 0 x...,,,,... x...,,0,,,... x...,,0,,,... x...,,,,... Range: y Range: y Vertical Asmyptotes : VerticalAsmyptotes : x...,,,,... x...,,0,,,... Graph at least one full period of each function. (State the asymptotes (if any) and period)... y tan x. g( x) cot( x) Per Aysmptotes : x,,... Per Aysmptotes : x,,,... Either use the formulas provided or use the basic shapes and shift your graph according to the new period. Boo method to find asymptotes for Tangent. cos( x) 0 x...,,,,... x...,,,,... x...,,,,... Boo method to find asymptotes for Cotangent. sin( x) 0 x,0,, x,0,, Set x = to the asymptotes for y tan x and y cot x and solve for x to find the new asymptotes. Page 4 of 9

5 Graphs of Secant and Cosecant y asec( x) y acsc( x) Amplitude a a Per ( see graph) Aysmptotes : x,,,... Amplitude a a Per ( see graph) Aysmptotes : x,0,,,... You don t necessarily need to memorize the formulas for asymptotes. Find out where the denominator will be zero and use the period of the function to find the other asymptotes. Secant and Cosecant Graphs a a y asec( x) y acsc( x) cos( x) sin( x) Period Period (n ) Domain:,except ; n n Domain:,except ; n cos( x) 0 sin( x) 0 x...,,,,... x...,,0,,,... x...,,0,,,... x...,,,,... Range: y a or y a Range: y a or y a Vertical Asmyptotes : VerticalAsmyptotes : x...,,,,... x...,,0,,,... Page 5 of 9

6 Graph at least one full period of each function. (State the asymptotes (if any) and period) Hint: Graph asin( x) or acos( x ) first. Then draw in the reciprocal values.. y sec x 4. f ( x) csc( x). y sec x 4. f ( x) csc( x) Amplitude Amplitude Per ( see graph) Per Aysmptotes : x,,,... Aysmptotes : x,0,,,... Either use the formulas provided or use the basic shapes and shift your graph according to the new period. Boo method to find asymptotes for Secant. cos( x) 0 x...,,,,... x...,,,,... Boo method to find asymptotes for Cosecant. sin( x) 0 x,0,, 0 x,,, x,0,, Just set denominator equal to zero. Once you find the first zero, add the period to get the next one and so on Section 4.7 Inverse Trigonometric Functions Defintions of Inverse Trig Functions Function Domain Range meansiff x values y values y sin ( x) sin y x [,], y cos ( x) cos y x [,] [0, ] y tan ( x) tan y x (, ), arcsin and sin Page 6 of 9 Loo at page 45. Remember this!!! Inverse trig functions are just angles. You are finding the angle that has a particular x or y value. Note: We could also use degrees., is 90 x 90 0, is 0 x 80, is 90x 90 are the same thing.

7 Evaluate the expression without a calculator.. arcsin Let arcsin, then sin. Therefore arcsin( ) 90 or HINT: Thin unit circle and domain of inverse function:, or 90x90 for arcsine.. cos () Let cos, then cos. Therefore cos () or80 0, or 0 x 80 for arccosine.. arctan( ) Let arctan, then tan. Therefore arctan( ) 45 4 or Must be in Q IV., or 90 x arccos Must be in Q II. Let cos, then cos. 5 Therefore cos 50 6 or 0, or 0 x 80 for arccosine. 5. tan Must be in Q I. Therefore tan or 0 6 sin Let tan, then tan. Thin 6 tan 6 cos 6 HINT:Use the unit circle. We are looing for the ratio of y x such that the ratio is equal to. This is going to occur either at at, or at,. Since the 6 tan opp y. 6 adj x Evaluate the expression without a calculator. is in the denominatorr, let s try 6 first. 6. Must be in Q IV. Therefore cos 0 6 or Let sin, then sin., or 90x90 for arcsine. sin Page 7 of 9

8 Evaluate the expression with a calculator. Write your answer in radians and degrees. Round the result to two decimal places. sin Radians: sin Degrees: 8. tan 8 sin.55.7 Know your calculator! Are you in Degrees or Radians?? Radians: tan 8.45 Degrees: tan arccos Radians: arccos.4 Degrees: arccos 80 Your calculator is programmed smartly. It will give you the correct answer but remember what mode you are in!! Loo at the screen every time to see if you are in degrees or radians. Most calculators display this setting on the home screen. (Or clic on mode) Section 4.8 Applications Solve the right triangle for unnown angles and sides. Round the result to two decimal places.. b =., c = 9.45 Answer Given: b =., c = 9.45 Find A,B, C, a adj b cos A hyp c.. cos A A cos A8.97, B 8.0 C 90, a 9.6 B Solve the right triangle for unnown angles and sides. Round the result to two decimal places.. B 75, b Answer Given: B 75, b A5, C 90 Find A, C, a, c a.0, c.4 a b c a c b a opp b sin B hyp c sin 75 c.4 c sin 75 A Find a next. a b c a c b a Page 8 of 9

9 . A 60, b 0 Answer Given: A 60, b 0 B 0, C 90 Find B, C, a, c a7., c0 adj b cos A hyp c 0 0 cos 60 c 0 c cos60 B Find a next. a b c a c b a Extra Credit problem Write an algebraic expression that is equivalent to the expression. (Use a triangle) x EC. sec(arctan( x )) Hint: x x x Remember, inverse trig functions are angles. Let tan, then tan. Draw triangle and use Pythagorean theorem to find the missing side. (opposite theta,, in this case) c ( x) 9x, c 9x Use triangle to find tan. hyp 9x sec 9x adj Done!!!! 4 EC. csctan Hint: Which quadrant is tan < 0? Answer: Q IV (+, ) = (cos, sin) 4 4 Remember, inverse trig functions are angles. Let tan, then tan. Draw triangle and use Pythagorean theorem to find the missing side. (opposite theta,, in this case) c 4 5, c 5 Use triangle to find tan. hyp 5 5 csc Done!!!! opp 4 4 Sine is negative in quadrant 4. Page 9 of 9