1)solve for x in the equation :sin(4x-10°)-cos(x+60°)=0
2)Given that cosA=0.64,without using mathematical table or calculator find;
a)sin A
b)Tan (90-A)
3)Two students Paul and Omondi standing 10m apart on the same side of a tall building on a horizontal ground.Paul who is closer to the building sees the roof top at an angle of 70° while Omondi at an angle of 46.8° if the building Paul and Omondi lies on a straight line calculate the height of the building to 3 significant figures
4)solve the following inequalities and state the integral solutions;1/2(24-4x)>6(3x-4/3)>\=-2/3(42+3x)
5)Given that the inequalities x-5<\=3x-8<2x-3;
a)solve the inequalities
b) represent the solution on a number line.
1) sin(4x-10°) - cos(x+60°) = 0
sin(4x-10°) = cos(x+60°)
sin(4x-10°) = sin(30°-x)
4x-10° = 30° - x (since both angles have the same sine)
5x = 40°
x = 8°
2) a) sin A = √(1-cos^2A) = √(1-0.64^2) ≈ 0.768
b) Tan(90-A) = 1/TanA = 1/√(1-cos^2A)/cosA = cosA/√(1-cos^2A) = 0.8/0.48 ≈ 1.667
3) We can form a right triangle with the height of the building as the hypotenuse, and the distances from Paul and Omondi to the building as the adjacent legs. Let h be the height of the building in meters, then we have:
h/tan70° + 10/tan46.8° = h
Solving for h:
h = (10/tan46.8°)/(1/tan70° - 1)
h ≈ 20.8 meters
4) 1/2(24-4x) > 6(3x-4/3) ≥ -2/3(42+3x)
12-2x > 18x - 8 ≥ -28 - 2x
14x < 20 < 20x
1 < x < 5/7
The integral solutions are x=1, 2, 3, 4.
5) a) x-5 ≤ 3x-8 < 2x-3
-x ≤ -3 < -x+2
-5 ≤ x < -2
b) On a number line, we can represent the solution as a closed interval between the endpoints -5 and -2, with a filled-in circle at -5 and an open circle at -2.
1) To solve the equation sin(4x-10°) - cos(x+60°) = 0:
Step 1: Use the identity sin(A) - cos(B) = sqrt(2)sin(A + 45° - B) to rewrite the equation:
sqrt(2)sin(4x - 10° + 45° - (x + 60°)) = 0
Step 2: Simplify the equation:
sin(3x - 25°) = 0
Step 3: Find the values of x that satisfy the equation:
sin(3x - 25°) = 0
This occurs when 3x - 25° is an integer multiple of 180°.
Case 1: 3x - 25° = 0
3x = 25°
x = 25°/3
Case 2: 3x - 25° = 180°
3x = 205°
x = 205°/3
The solutions are x = 25°/3 and x = 205°/3.
2) To find sin(A) and tan(90-A) given cos(A) = 0.64:
Step 1: Use the identity sin^2(A) + cos^2(A) = 1 to find sin(A):
sin^2(A) + 0.64^2 = 1
sin^2(A) = 1 - 0.64^2
sin(A) = sqrt(1 - 0.64^2)
Step 2: Use the identity tan(90-A) = cot(A) to find tan(90-A):
tan(90-A) = 1/tan(A)
tan(90-A) = 1/(sqrt(1 - 0.64^2)/0.64)
a) sin(A) = sqrt(1 - 0.64^2)
b) tan(90-A) = 1/(sqrt(1 - 0.64^2)/0.64)
3) To calculate the height of the building given the angles of elevation:
Step 1: Use the tangent function:
tan(70°) = height/base
tan(70°) = h/10
Step 2: Use the value of h from the previous equation in the second equation:
tan(46.8°) = (h + 10)/10
Step 3: Set up the equations and solve for h:
tan(70°) = h/10
tan(46.8°) = (h + 10)/10
Solve these equations simultaneously to find the value of h. Round the answer to 3 significant figures.
4) To solve the inequality equation 1/2(24-4x) > 6(3x-4/3) >= -2/3(42+3x):
Step 1: Simplify the equation:
12 - 2x > 18x - 8 >= -28 - 2x
Step 2: Solve each part of the inequality separately:
12 - 2x > 18x - 8
10x < 20
x < 2
18x - 8 >= -28 - 2x
20x >= -20
x >= -1
The integral solutions are -1 < x < 2.
5) To solve the inequalities x-5 <= 3x-8 < 2x-3:
Step 1: Solve the first inequality:
x - 5 <= 3x - 8
3x - x >= 3
2x >= 3
x >= 3/2
Step 2: Solve the second inequality:
3x - 8 < 2x - 3
3x - 2x < 8 - 3
x < 5
a) The solution is 3/2 <= x < 5.
b) Represent the solution on a number line:
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0 3/2 5