| 1 | Find exact value | sin(30) | |
| 2 | Find exact value | sin(45) | |
| 3 | Find exact value | sin(30 deg. ) | |
| 4 | Find exact value | sin(60 deg. ) | |
| 5 | Find exact value | tan (30 degrees) | |
| 6 | Find exact value | arcsin(-1) | |
| 7 | Find exact value | sin(pi/6) | |
| 8 | cos(pi/4) | ||
| 9 | Find exact value | sin(45 deg. ) | |
| 10 | Find exact value | sin(pi/3) | |
| 11 | Find exact value | arctan(-1) | |
| 12 | Find exact value | cos(45 deg. ) | |
| 13 | Find exact value | cos(30 deg. ) | |
| 14 | Find exact value | tan(60) | |
| 15 | Find exact value | csc(45 deg. ) | |
| 16 | Find exact value | tan (60 degrees) | |
| 17 | Find exact value | sec(30 grad. ) | |
| 18 | Find exact value | cos(60 deg. ) | |
| 19 | Find exact value | cos(150) | |
| 20 | Find exact value | sin(60) | |
| 21 | Find exact value | cos(pi/2) | |
| 22 | Find exact value | tan (45 degrees) | |
| 23 | Find exact value | arctan(- square root of 3) | |
| 24 | Find exact value | csc(60 deg. ) | |
| 25 | Find exact value | sec(45 grad. ) | |
| 26 | Find exact value | csc(30 deg. ) | |
| 27 | Find exact value | sin(0) | |
| 28 | Find exact value | sin(120) | |
| 29 | Find exact value | cos(90) | |
| 30 | Convert from radians to degrees | pi/3 | |
| 31 | Find exact value | tan(30) | |
| 32 | Convert from degrees to radians | 45 | |
| 33 | Find exact value | cos(45) | |
| 34 | Simplify | sin(theta)^2+cos(theta)^2 | |
| 35 | Convert from radians to degrees | pi/6 | |
| 36 | Find exact value | cot(30 grad. ) | |
| 37 | Find exact value | arccos(-1) | |
| 38 | Find exact value | arctan(0) | |
| 39 | Find exact value | cot(60 deg. ) | |
| 40 | Convert from degrees to radians | 30 | |
| 41 | Convert from radians to degrees | (2pi)/3 | |
| 42 | Find exact value | sin((5pi)/3) | |
| 43 | Find exact value | sin((3pi)/4) | |
| 44 | Find exact value | tan(pi/2) | |
| 45 | Find exact value | sin(300) | |
| 46 | Find exact value | cos(30) | |
| 47 | Find exact value | cos(60) | |
| 48 | Find exact value | cos(0) | |
| 49 | Find exact value | cos(135) | |
| 50 | Find exact value | cos((5pi)/3) | |
| 51 | Find exact value | cos(210) | |
| 52 | Find exact value | sec(60 grad. ) | |
| 53 | Find exact value | sin(300 deg. ) | |
| 54 | Convert from degrees to radians | 135 | |
| 55 | Convert from degrees to radians | 150 | |
| 56 | Convert from radians to degrees | (5pi)/6 | |
| 57 | Convert from radians to degrees | (5pi)/3 | |
| 58 | Convert from degrees to radians | 89 city. | |
| 59 | Convert from degrees to radians | 60 | |
| 60 | Find exact value | sin(135 deg. ) | |
| 61 | Find exact value | sin(150) | |
| 62 | Find exact value | sin(240 deg. ) | |
| 63 | Find exact value | cot(45 deg. ) | |
| 64 | Convert from radians to degrees | (5pi)/4 | |
| 65 | Find exact value | sin(225) | |
| 66 | Find exact value | sin(240) | |
| 67 | Find exact value | cos(150 deg. ) | |
| 68 | Find exact value | tan(45) | |
| 69 | Calculate | sin(30 deg. ) | |
| 70 | Find exact value | sec(0) | |
| 71 | Find exact value | cos((5pi)/6) | |
| 72 | Find exact value | csc(30) | |
| 73 | Find exact value | arcsin((square root of 2)/2) | |
| 74 | Find exact value | tan((5pi)/3) | |
| 75 | Find exact value | tan(0) | |
| 76 | Calculate | sin(60 deg. ) | |
| 77 | Find exact value | arctan(-( square root of 3)/3) | |
| 78 | Convert from radians to degrees | (3pi)/4 | |
| 79 | Find exact value | sin((7pi)/4) | |
| 80 | Find exact value | arcsin(-1/2) | |
| 81 | Find exact value | sin((4pi)/3) | |
| 82 | Find exact value | csc(45) | |
| 83 | Simplify | arctan( square root of 3) | |
| 84 | Find exact value | sin(135) | |
| 85 | Find exact value | sin(105) | |
| 86 | Find exact value | sin(150 deg. ) | |
| 87 | Find exact value | sin((2pi)/3) | |
| 88 | Find exact value | tan((2pi)/3) | |
| 89 | Convert from radians to degrees | pi/4 | |
| 90 | Find exact value | sin(pi/2) | |
| 91 | Find exact value | sec(45) | |
| 92 | Find exact value | cos((5pi)/4) | |
| 93 | Find exact value | cos((7pi)/6) | |
| 94 | Find exact value | arcsin(0) | |
| 95 | Find exact value | sin(120 deg. ) | |
| 96 | Find exact value | tan((7pi)/6) | |
| 97 | Find exact value | cos(270) | |
| 98 | Find exact value | sin((7pi)/6) | |
| 99 | Find exact value | arcsin(-( square root of 2)/2) | |
| 100 | Convert from degrees to radians | 88 city. |
Test, grade 10
Final control in algebra and the beginnings of analysis
Test grade 10
Teacher N.P. Gryzina MBOU School No. 3, Ozyory.
Boption 1.
Part one.
1. Simplify the expression: (2sin2x – 2cos2x)tg2x
1. –cos2x. 2. 2. 3. – 4. 4. – 2sin2x
2.Find the value of the expression: 6tg2x - 2 if cos2x = 0.5
– 2. 2. – 5. 3. 22.4. 4.
3.Calculate: sin550 cos350 + cos550 sin350.
1. 1. 2.0. 3.Sin200. 4.— 2
4.Find the set of values of the function y = 3cos28x - 2.
2.3.4.
5. Solve the equation: sinx - = 0
1.+2πn, nZ. 2. (- 1)n + , nZ
3.+n, nZ. 4.± +πn, nZ.
6.Solve inequality: ≥0
[2;+).2. (- ; — 4)
3.( — 4; — ] [2; +).4. (-;
7.The function is given by a graph on the interval . Indicate the values of x for which the function is negative.
2. 3. (-1; 2).
4.[-4; -3) (-1; 2).
8.Find the derivative of the function y \u003d 4 × 3 - 2cosx
1. 12×2 + 2sinx. 2.12×3 + 2sinx. 3. 7×2 – 2sinx. 4.3×2 – 2cosx.
9.Specify an even function.
1. y=sinx – x2. 2. Y=x2 + x + cosx.3.Y=sin2x + x3.4. Y=7×2 +cos3x.
10. Find the slope of the tangent drawn to the graph of the function y \u003d x2 at the point with the abscissa x0 \u003d 1.
1.1. 2.2. 3. 3. 4. 0,5.
IN 1.Find the value of the expression: tg(π+α)sin( – α)cos( +α) for α=–
AT 2.The point moves in a straight line according to the law x(t) = 2t3 - 1.5t2 + 5 (where t is the time in seconds, x is the distance in meters). Calculate the speed of the point at time t = 2c.
B3.How many integers are in the scope of the function
F(x) = ?
Part two.
AT 4.The function is defined on the segment .
The figure shows a graph of its derivative. Specify the number of minimum points of the function y = f(x).
B5.Find the longest interval of decreasing function
y= x3– 4×2 + 6x +3.
B6.Determine the number of roots of the equation 2sin2x - 3sinx - 2 = 0 on the segment.
AT 7.Find f(x0) if f(x) = (3x–5)2 + , x0= 2.
C1.Find the set of values of the function f(x) = x+cos2x defined on the interval .
With 2. Find all solutions of the system of equations
satisfying the condition (x - y).
With 3.Find all values of the parameter p for which the given equation has no roots:
4sin3x+3cos2x+p=0.
Test grade 10
Option 2.
Part one.
A 1.Simplify the expression: sin22x.
1.–1.2.2. 3. 0. 4. 4.
A2. Find the value of the expression: 7 - 3cos2x if tg2x = 2.
1. 1.2.6.3. 5,5.4.4.
A 3. Calculate: cos 700 cos 200 - sin 700sin 200 .
1.–1. 2. Sin500. 3. Cos500. 4. 0.
A4. Find the set of values of the function y = 6 + sin2x.
1. . 2..3. .4..
A5.Solve the equation: sin =.
1.(–1)n2. ±.3. (–1)n4.±
A6.Solve the inequality:
1.( –4;2][4;+).2.(–3. (–4.[–4;2]
A7.The function is given by a graph on the interval . Indicate those x values for which the function is positive.
1.(–3;–1)2.(0;3].3.[–3;–1].4. (3;4).
A8. Find the derivative of a function: y= 6x4 - 3
1. 10x+3cosx. 2. 24×4– 3cosx. 3.24×3- 3cosx.4.4×3+3sinx.
A9.Specify an odd function.
1. y=x7+cosx.
2.Y=x5+2sinx.3. Y=2×3–cos2x.4.Y=x4+sinx.
A10.Find the slope of the tangent drawn to the graph of the function y \u003d 5x3–7x at the point with the abscissa x0 \u003d 2.
1.23. 2. 67. 3.8.4.53.
IN 1. Find the value of the expression: tgsin(2π-α)cos(π+α), with α=.
AT 2.The body moves in a straight line according to the law x(t)=3t3 - 2t2 - t (where t is the time in seconds, x is the distance in meters). Calculate the speed of the body at the moment t=2c.
B3.How many integers are in the scope of the function
F(x)=.
Part two.
AT 4.The function is defined on the interval [-5;3]. The figure shows a graph of its derivative. specify the number of maximum points of the function y=f(x).
B 5. Find the longest interval of increasing function
F(x)=–.
B 6. Determine the number of roots of the equation on a given interval:
2cos2x–5cosx+2=0, [0;π].
B 7. Find f'(x0) if f(x)=.
C 1. Find the set of values of the function f(x)= x – , defined on the interval .
C 2. Find all solutions of the system of equations that satisfy the condition (x + y).
C 3. Find all values of the parameter p for which this equation has at least one root pctg2x+2sinx+p=3.
Test Grade 8
Option 1.
Part one.
1.Represent the number - 0.125 as a square or cube.
A. (–0,25)2. B.(–0,5)3. IN. (–0,25)3.D. It is impossible to imagine.
2.Expressions are given: 1) ; 2) ; 3) Which of these expressions do not make sense at x=3?
A. only 2. B. only 1. IN.1 and 3. D.1 and 2.
3.Simplify the expression:
Answer:____________
4.What is the value of the expression ?
A.
5.Solve an equation: 7x2+9x+2=0.
A.
no roots. B.7; –2. IN. –1;. D. ; 1.
6.Find the value of an expression:
Answer:_______________
7.Solve the inequality: 5x+1
A.(–B.(2;+.IN. (–D.(–2;+
8.Solve the equation: x2+3x=0.
A.0;3. B. 0;–3.IN.0. D.–3.
9.Sort the numbers in ascending order.
Answer:________________
10.Solve the system of inequalities:
Answer:_________________
11. Which of the following statements is true about the equation:
-3x2=2–x?
A.The equation has one root. B. The equation has no roots.
IN.has two roots of different signs. D. has two roots of the same sign.
12. For each graph, specify the function corresponding to it:
A) y=x2;B) u=–1.5x2; IN) y=-2x+2; D)in=
Part two.
1. (2 points) Solve the equation:
.
2.(4 points) Solve the system of inequalities:
3. (6 points) The boat travels 8 km upstream and another 30 km downstream in the same time it takes a raft to travel 4 km in this river. The speed of the boat in still water is 18 km/h. Find the speed of the river.
Test Grade 8
Option 2.
Part one.
1.Represent the number 0.0027 as a square or cube.
A.(0,09)2. B.0,33. IN.(0,03)2. D.It is impossible to imagine.
2. Expressions are given: 1) ; 2) Which of these expressions do not make sense at x=-3.
A.Only 1. B. 1 and 2. IN. 2 and 3.D.1 and 3.
3.Simplify the expression:
Answer:__________
4.What is the value of the expression ?
A.. B.. IN. –25. D.25.
5.Solve the equation: 5x2–7x+2=0.
A.1,6; –. B. 1; .IN. –1;–0,4.D.There are no roots.
6.What is the value of the expression
Answer___________
7. Solve the inequality: 2x + 5.
A. (–B.(–IN.(–1;+D. (4;+
8.Solve the equation: x2-9x=0.
A.–9. B.0; 9. IN.0.D. 0;–9.
9.Arrange the numbers 2.5; and in ascending order.
Answer:____________
10. Solve the system of inequalities:
Answer:____________
11.
Which of the following statements is correct regarding the equation 3x2=4-x?
A.The equation has a single positive root.
B.The equation has a single negative root.
IN.The equation has two roots of different signs.
D.The equation has two roots of the same sign.
12. For each graph, specify the function corresponding to it.
A.u=1.5x2;B. in=; IN.Y=2x; D.Y=-x3;
Part two.
1. (2 points) Solve the equation: – =.
2. (4 points) Solve the system of equations:
3.
(6 points) The motorcyclist covered the distance from point M to point N in 5 hours. On the way back, he rode the first 36 km at the same speed, and the rest of the way at a speed of 3 km/h more. At what speed did the motorcyclist initially drive if he spent 15 minutes less on the way back than on the way from M to N?
sinx + cosx = sqrt(2) cos2x
CHHAYA PUBLICATION - GENERAL SOLUTIONS OF TRIGNOMETRIC EQUATIONS - Short Answer Questions
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Updated on: 27-06-2022
Text solution
Answer
Nπ-π4,2nπ+π12,2nπ-7π12
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sin theta +sin 2theta+sin3theta+sin4theta=0 92x_1`
`sinx_1=sqrt(2)`
If you try to calculate `arcsinsqrt2` in your calculator, you will get an error. This is because it doesn't exist. The values that exist for arcsin range from -1 to 1, and `sqrt(2)>1`
The value of `sin((3pi)/4)` is `sqrt(2)/2`; hence it cannot be a solution for `arcsinsqrt2` . Rather, it is the product of one existing root `pi/4`.
Also note in the graph of the function that the graph does not intersect the x-axis at sqrt(2), further confirming that the root does not exist. Rather, all roots of this function are multiples of 1 existing root, pi/4, since this is an infinitely repeating function. 92x_2`
Because you can't take the square root of a negative number, x2 doesn't exist.
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